Book flight without credit card

The rules on how long flight reservations may be held without ticketing will depend on the underlying fare. In my experience of public fares, the limit is between three days and a week, except in the case of Qantas.

It is not the decision of the travel agent, but rather the airline. Ticketing will not occur until payment is made, but confirmed reservations can be made without payment, and a PNR created to hold them. In the days of paper tickets, this was the standard procedure. These days it is less common to do it separately but it is certainly possible.

I have not used the site you are talking about, but I regularly set up PNRs with quoted and confirmed flight reservations and pay for them separately at a later time. I do this largely because my itineraries are too complicated to do online, but I have a reliable and talented contact at the airline who accepts PNR requests by email from me.

Some private fares, for instance those offered by corporate travel agents, or to airline employees, will be much more generous, perhaps not requiring ticketing until the time of departure.

I include some examples. For instance, the very cheapest BA fare on London-Rio de Janeiro includes the following rule.

ADVANCE RESERVATIONS/TICKETING

 CONFIRMED RESERVATIONS FOR ALL SECTORS ARE REQUIRED AT
   LEAST 7 DAYS BEFORE DEPARTURE.
   TICKETING MUST BE COMPLETED WITHIN 3 DAYS AFTER
   RESERVATIONS ARE MADE OR AT LEAST 7 DAYS BEFORE
   DEPARTURE WHICHEVER IS EARLIER.

If you fail to adhere to this rule, a process called the "ticketing robot" (which is just a computer program) will come through and delete any reservations if they persist after the ticketing time limit on the PNR. I have never seen a more generous rule than three days on BA or on American Airlines internationally.

That said, many domestic American Airlines fares are much more rigid. This is the cheapest fare on Rochester-Dallas:

ADVANCE RESERVATIONS/TICKETING

 CONFIRMED RESERVATIONS FOR ALL SECTORS ARE REQUIRED AT
   LEAST 21 DAYS BEFORE DEPARTURE ON EACH TRIP.
   WAITLIST NOT PERMITTED.
   TICKETING MUST BE COMPLETED BY MIDNIGHT 1 DAY AFTER
   RESERVATIONS ARE MADE OR AT LEAST 21 DAYS BEFORE
   DEPARTURE WHICHEVER IS EARLIER.

The most expensive first class fares on the same route only gives maximum three days:

ADVANCE RESERVATIONS/TICKETING

       NOTE - TEXT BELOW NOT VALIDATED FOR AUTOPRICING.
          RESERVATIONS BOOKED MORE THAN 4 DAYS PRIOR TO DEPT
          MUST BE TICKETED AT LEAST 3 DAYS DEFORE DEPT.
          RESERVATIONS BOOKED 3 DAYS OR LESS PRIOR TO DEPT
          MUST BE TICKETED WITHIN 24 HOURS. RESERVATIONS
          BOOKED 24 HOURS OR LESS PRIOR TO DEPARTURE MUST BE
          TICKETED WITHIN 3 HOURS.

The standard discount Qantas domestic business fare includes the following provision, which essentially allows unlimited holding up until five hours prior to departure time.

ADVANCE RESERVATIONS/TICKETING

   CONFIRMED RESERVATIONS ARE REQUIRED FOR ALL SECTORS.
   WAITLIST NOT PERMITTED, TICKETING MUST BE COMPLETED AT
   LEAST 5 HOURS BEFORE DEPARTURE.
   OR - CONFIRMED RESERVATIONS FOR ALL SECTORS AND
        TICKETING MUST BE COMPLETED AT THE SAME TIME.
        WAITLIST NOT PERMITTED.

(With less than five hours to departure, ticketing must be completed immediately.)

Qatar Airways (QR) is also a bit more generous, giving you a week on the outside.

   GENERAL RULE - APPLY UNLESS OTHERWISE SPECIFIED
         NOTE - TEXT BELOW NOT VALIDATED FOR AUTOPRICING.
          QR TTL NOTE-
          WHEN RESERVATIONS ARE MADE AT LEAST 11 DAYS BEFORE
          DEPARTURE TICKETING MUST BE COMPLETED WITHIN
          8 DAYS AFTER RESERVATIONS ARE MADE OR AT LEAST
          7 DAYS BEFORE DEPARTURE WHICHEVER IS EARLIER.
          --------------------------------------------------
          WHEN RESERVATIONS ARE MADE AT LEAST 4 DAYS BEFORE
          DEPARTURE TICKETING MUST BE COMPLETED AT LEAST 72
          HOURS BEFORE DEPARTURE.
          --------------------------------------------------
          WHEN RESERVATIONS ARE MADE AT LEAST 2 DAYS BEFORE
          DEPARTURE TICKETING MUST BE COMPLETED AT LEAST 24
          HOURS BEFORE DEPARTURE.
          --------------------------------------------------
          RESERVATIONS AND TICKETING MUST BE COMPLETED AT
          THE SAME TIME.

If combining several fares on one ticket, the most restrictive fare of them all will generally apply; although the confirmed flights may not all disappear at once. The only exception is that some very special fares such as round-the-world or visit-a-continent fares may supersede ticketing time limits when combined with more restrictive fares. That's because they must be manually priced, a process requiring many man hours.

Most airlines that I use require ticketing to be completed at least one hour before departure, notwithstanding the deadlines above. With less notice, they will require reservations and ticketing to be done simultaneously, if they are willing to do it at all.

As a word of warning, I have also found that the Iberia robot is very aggressive and will delete held sectors prior to the PNR expiry date. You do not have a contract with the airline until you have exchanged money, and they will not regard a contract as existing until the ticket is issued. If they decide to pull or edit the fare prior to ticketing, it may (or may not) invalidate your fare quote under the PNR. So you should not rely too much on confirmed reservations until issuance occurs.

As a comment, I am going to say I place a lot of value on being able to create PNRs and pay for them later. However, when you create a confirmed reservation, it pulls the seat out of the inventory and the seat appears to be "sold" to the computer system, even if there is no ticket sale yet. This will cause prices to go up for other travellers, and if you do not purchase the seat, it is essentially revenue spoilage for the airline. I don't ticket all of the PNRs I create, but I would ask that you not violate the usefulness of this system by creating PNRs you never intend to ticket. This might cause airlines to clamp down on the procedure and tighten the deadlines.

Finally, I will also add that the reason you don't see many online travel agents selling these "reserve now, pay later" at no risk to themselves, is the airlines will consider the creation of a large number of unticketed PNRs to be a violation of the travel agent's ticketing authority. As I say, it distorts the pricing system to pull seats out of inventory and then drop them back in, so the airlines discourage this practise and in some cases, such as QR, will start fining the travel agent for doing this.