Firstly, I doubt very much whether the German Federal Constitutional Court (Bundesverfassungsgericht) would have been fooled by the ruse for an instant! The legality of such a move would almost certainly have been challenged.

As Pieter Geerkens has pointed out in the comments,

... the whole point of a constitution is to make work-arounds difficult, ensuring time for lengthy consultation with multiple experts and political groups. It is intended to be a framework document, not merely legal restriction.

Perhaps more importantly from a practical standpoint, in this instance the hostage-taking and subsequent massacre at the 1972 Munich Summer Olympics (which is the context for the quote) occurred over a period of just 2 days (5 - 6 September 1972). As anyone who has served in the military will attest, it takes much, much longer than that to resign from the army!

A bit on the counterfactual side as this is, here are some small points to consider, most shall not contradict sempaiscuba's or Schwern's answers:

1. The whole affair started in the early morning hours of September 5th, 1972 and was essentially over the next day. In a situation of extremely high tension, with wholly unexpecting and generally unprepared officials and organisational structures that are now seen as completely ineffective and unsuitable: that is quite a short time to react.

2. As explained in the Wikipedia article, one, if not the main concern publicity wise was that these were supposed to be "friendly" games, in a "changed Germany" (no, I will not mention the war.) This attitude lead to the deployment of mostly unarmed and even largely un-uniformed police during the games. And just as "nobody expects the Spanish Inquisition" nobody thought of bad guys turning up at such a type of event, ever.

3. Using the army "for internal affairs" was indeed strongly prohibited at the time by the basic law. Having specialists from the military resign and sign up immediately with the police would be quite a "dirty trick", manoeuvring like this would have been time consuming as well as unattractive for anyone involved.

But:

1. In 1962 during the North Sea Flooding the then police senator¹ of Hamburg (Helmut Schmidt) did violate the constitution and did request emergency help from the military for "internal affairs". He was not punished, not reprimanded. He was lauded with respect for daring to technically violate the constitution. It caused some debates, but in 1968 the law was changed to explicitly exclude disaster relief from being prohibited.

2. Communications were not that inefficient as now commonly portrayed. During the cold war West-Germany was well prepared for alerting its military and other emergency structures on short notice. This was improved with every catastrophe, the mentioned flood being one of those.
   This is evidenced by the fact that not only the local Munich police got its hands in the action. Both the cabinets of Israel and West-Germany deliberated about the crisis, a special crisis committee (Krisenstab) of the German government was created (headed by: Federal Minister of the Interior, Bavarian State Minister of the Interior, police president, presidents of the National Olympic Committee and International Olympic Committee, and a State-secretary). These were assembled by the use of telephones and radios. Quickly. The whole structure of the crisis committee was set up by 10:00.

Quite apart from the delicate hierarchy of city, Land, and state, a “tangled web of responsibilities and competences,” as Matthias Dahlke recently put it, developed, hindering effective decision-making.[…]
This melee and the unreality of the situation it sought to master produced a number of impractical suggestions as to how the hostages might be liberated. These give a good insight into the psychological state of those who made them or listened to them. Brundage wasted valuable time having the crisis committee investigate knockout gas, which he erroneously remembered being used by the Chicago police in the 1920s to overpower the mob. The group also received fantastical schemes sent in by members of the public: a Bremen resident and Knight’s Cross holder from the Second World War offered to free the hostages in a surprise attack; a man from Landau proposed to go in with local friends; and another wanted to “overwhelm” the hostage-takers with one hundred thousand demonstrators.
From the outset, military solutions were considered too dangerous, although contingency plans had been put in place should the terrorists begin to execute the hostages or try to escape. From late afternoon, three attempts were made or considered. Shortly beyond the last ultimatum at 5:00 P.M., the police decided to storm the building with over three dozen officers. But the operation had to be abandoned since, due to a failure to order a news blackout, the terrorists were able to join the worldwide television audience in watching inexperienced volunteers in tracksuits dropping equipment and ammunition on the gravel roof above their heads. A second plan to shoot the Palestinians as they walked to the helicopters waiting to take them to the plane ended in an equally undistinguished manner, when the group’s leader who decided to walk the underground route first was alerted to the presence of machine-gunners and precision marksmen. And finally a farrago of errors at Fürstenfeldbruck military airport twenty kilometers to the north of the site brought Black September’s ultimate sanction and Schreiber’s worst fear to realization.
Untrained in close combat and fearing for their lives, a special commando of twelve volunteer police officers, posing as flight assistants on the waiting Lufthansa plane, abandoned their mission to overpower the terrorists shortly before the helicopters bringing them to the airport landed. When a shootout started, police marksmen were ill-positioned, badly lit, and scandalously under-equipped; support from armored vehicles took an age to weave through the heavy traffic and onlookers clogging the city center. After the lengthy gun battle claimed the lives of the nine remaining hostages, five of the eight terrorists, and Munich police brigadier Anton Fliegerbauer, in addition to leaving one of the helicopter pilots seriously injured through friendly fire, government spokesman Conrad Ahlers made a late-night announcement—based incredibly on a mysterious, unknown source—that the Israelis’ lives had been saved. Newspaper first editions around the world rushed to publish a happy ending its readers would already know to be inaccurate.
Two systemic flaws outweighed the basic negligence: the police sent too few marksmen, and those they did were reluctant to kill. Quite apart from failing to ascertain the number of terrorists until inexplicably late (and then passing on the information incorrectly), the police chose to operate with an inaccurate one-to-one ratio of marksmen to terrorists. The previous year, a siege at the Deutsche Bank in Munich’s Prinzregentenstraße had been resolved by overwhelming the perpetrators three to one, but the police had been criticized when a hostage was believed to have died as a result of friendly fire. Statements by Schreiber and Wolf after the Olympic debacle suggest the earlier tragedy had led to excessive caution. More importantly, the marksmen’s training had prepared them to incapacitate but not eliminate their targets. Despite being told they were acting under emergency legal cover (Nothilferecht), one later admitted to having scruples (Beißhemmung) about taking the Palestinians’ lives. Psychological fears, individual and collective, contributed as much to the disaster as practical errors.
_{From: Kay Schiller & Christopher Young: "The 1972 Munich Olympics and the Making of Modern Germany", University of California Press: Berkeley, Los Angeles, 2010, p 197-201.}

That means:

Yes, the police tasked with dealing with the situation were not ready for this. But regarding the military trained snipers from the original question this also means that no jumping through loop holes would be required. As long as someone along the lines of communication with a degree of authority would have stood up and dared to request the military, that would have worked regarding the timing, the authority and the legal repercussions would likely have been minor, as seen by precedence from 1962. Of course, if things had turned out better in 1972 then they did. That is a big "if" in politics. As seen by the distribution of members of the crisis committee, politics and outward appearances were not to be underestimated at that time. Even militarily trained snipers were and are not a guarantee to success.

Even letting aside this about the real military, Bundeswehr, one often cited consequence of this chain of events is the creation of the GSG 9. That GSG means Grenzschutzgruppe, or Border Police. These units were formed before the re-armament of the Federal Republic to circumvent the "no-military" stance, then official. It was the only (quasi-)military of West-Germany before the Bundeswehr was created and was only renamed and changed roles to Federal Police in recent years. The GSGs retained their military combatant status until 1994!
So, before 1972s formation of the GSG 9: there were already 8 small highly trained military units available, legally to be used in internal/domestic affairs! They were just not requested by anyone!

Therefore the premise for the original question would have to be answered as: quite possibly these "snipers" would have been unneeded and useless anyway, rendering the what-if scenario of resign-&-signup superfluous.
While most decision makers made the impression of reacting slowly they were quite busy, not only behind the scenes. But with most hopes of solving this peacefully – by negotiation or persuasion – being crushed one after the other, the options to choose from became fewer and fewer.

Conclusion

This a mixture of unpreparedness for a very chaotic and hectic situation, incompetence and failures to properly communicate in unclear and unforeseen situations. A chain of haphazard and ad-hoc decisions, many tries and even more aborts. Not everyone failed, but the whole situation was a mess. Using the police was not an error per se, a kind of militarily trained marksmen were legally available, but any military would not have ensured a positive outcome. In the scenario as it unfolded there was no single point of failure.
Theoretically, everything needed would have been available or even was in place. It still ended in tragedy, not because of inevitability but almost understandably so. In hindsight.

1: "Police senator" was the position Helmut Schmidt held at the time of the flood. This was later reformed into the position of senator for the interior. That naming is so peculiar because Hamburg is a (hanseatic) city, and a federal state at the same time. It is equivalent to a minister of the interior for the state of Hamburg.

But why didn't they ask some snipers to resign from the armed forces and then hire them in the police force and deploy them?

In a Hollywood movie the police would call up the local army base who would have a crack team of snipers just sitting around. Some weaselly lawyer would point out it's illegal until some clever grunt suggests "why don't we just resign?" which they would at the drop of a hat. The snipers would race over with their sniper rifles, salute a few times, and take up positions. The kidnappers would all be killed. Heroic music plays. Roll credits.

Reality doesn't work like that. It moves a lot slower and clever legal tricks don't work.

Legality

The plan would likely be highly illegal.

Intent is very important in law. Technicalities don't save you if it can be shown you intended to violate the law. A court would easily show the Munich Police's intent to use the military on domestic soil both through official communications and timing of their resignations.

A law is not just its text, it is also its intent and case law. A judge looking at the ban on using the military domestically would likely object to the technicality of having them resign first violates the intent of the law.

Finally, there would also be the issue of the precedent they're setting. Laws, let alone Constitutional Laws, are not something you simply set aside in a crisis. If the police can ignore the Basic Law because there's a bad situation they weren't prepared for, this would set a very bad precedent for future police violations of Basic Law. Perhaps they could get special dispensation just this one time, but as we'll see below, they didn't have the time.

Morality

Western democracies are built on separation of powers, and they have very, very good reasons to keep the military separate from domestic affairs lest they take over. Particularly Germany who lived through the militarization of their Republic just 30 years prior. It's highly unlikely the Munich Police would have considered it morally acceptable to bring in the military on a technicality.

Just Six Hours

The speed at which this developed, the chaos and changing plans, and the breakdown of communications. Between 4:30am when the hostages were taken and 10:30pm when they were being moved to their helicopters is just 18 hours. This is not a lot of time in 1972.

But really they had less than 6 hours to arrange the airport ambush.

The original plan was to have a squad of police storm the compound by crawling through roof vents. At 4:30pm they entered the compound. The operation was blown when news crews broadcast them on live TV.

After this, plus getting frustrated by the negotiations, the kidnappers now demand to be flown to Cairo. The police saw this as an opportunity to ambush the kidnappers while they're transferring to the plane in the open spaces of an airport. They had to put this all together in less than six hours.

The New Plan Goes South

The new plan was to have the kidnappers and hostages fly in helicopters to the airport and board a waiting 727. It was expected that two kidnappers would inspect the plane where they'd be overpowered by German police waiting inside. Snipers would then take out the remaining kidnappers in the helicopters, expected to be just two or three.

It turns out there were eight kidnappers.

To make matters worse, at the last minute the police aboard the plane voted to abandon their mission, but didn't inform central command. The kidnappers find the aircraft empty and realized it's a trap!

Only now did it all fall on five untrained snipers to take out eight kidnappers.

No Cell Phones, No Internet, No Liaison

But let's say they did decide to use military snipers. What would they have had to do in the six hours between Operation Sunshine failing and cooking up the new airport ambush?

Something easily forgotten is how hard it was in 1972 to not only get in touch with somebody, but to find out how to get in touch with them, or who you're supposed to get in touch with. They'd be using personal contacts, phone books, and land lines.

The Munich Police had no reason to have an army liaison, they'd have to start cold. Who do they call? Once they figure that out, how do they get in touch with them? Whomever they contact would likely need to get authorization for such an unusual request; who do they call? Those officers would have to deliberate about such an unusual request, legally and morally. And so on.

Assuming they get the legal authorization for this plan, now the military has to not only find a team of readily available snipers. Snipers were not nearly as common in 1972 as they are now.

Not just any snipers. They'd have to find ones who are willing to resign from their career in the military on a moment's notice. Probably resign permanently because it would really play havoc with the law if they could just reenlist. If not already on base, they'd have to be found and recalled. Then do all the paper work to make the resignation official.

Then get them from wherever they are to Munich. If they're in, say, Frankfurt that's 4 hours by car.

Snipers vs Designated Marksmen

As a brief aside, most folks mix up Designated Marksmen and Snipers.

Snipers are highly specialized strategic assets. They're trained to operate alone (with a spotter) using camouflage, concealment, and fieldcraft to stay hidden. They provide intelligence and, if necessary, kill high-value targets. They often use specialized rifles, ammunition, and equipment. They operate independently and answer to higher levels of command, such as a battalion.

Designated Marksman are a normal member of an infantry squad who has received additional marksman training. They operate with the squad as a flexible asset for the squad commander, but can still fill all the normal infantry roles. A designated marksman's weapon will be optimized for longer range than normal (300 to 800 meters), but still use the same ammunition as everyone else.

Munich required people trained to operate within a larger group under strict command and control. Once the firing started things would rapidly get chaotic, so they'd also need to operate as infantry. As such, a designated marksman would be more appropriate than a sniper.

Integrating Outsiders

Ok, we have our snipers marksmen.

Now these two groups who have never worked nor trained together have to pull off a tense rescue operation on an extremely tight time scale.

Because they're no longer military they probably couldn't have brought their military equipment, so they need to be equipped. They'll have to familiarize themselves with Munich Police radios. The Munich Police have no special marksmen equipment, so they'll have to use the normal G3 battle rifle with no scopes hampering their ability as a marksmen. But at least it's the rifle they're familiar with.

Once that's done they'll have to be familiarized with the Munich Police's procedures, people, communications, and command. And the Munich Police command would have to be briefed on the capabilities and limitations of these army snipers in order to incorporate them into their plan. And they'd need to be in position and out of sight well before the kidnappers arrived at the airport.

GSG 9

This is all why Germany formed GSG 9. They couldn't use the army. Even if they could, they couldn't slap military and police units with different equipment and procedures who have never trained together in a few hours and expect them to succeed in tense, precision operations.