Did modern soldiers try to miss their shots?

Pretty broad question, so many cultures, countries and wars to consider. But here is one set of statistics, from an article here.

Russell W. Glenn, the author of book Reading Athena’s Dance Card: Men Against Fire in Vietnam,

conducted a survey of 258 1st Cavalry Division Vietnam veterans in 1987

his findings showed that

Only nine of the 1st Cavalry Division veterans reported that they never personally fired on the enemy

and when asked to estimate fellow soldiers actions:

veterans were therefore also asked to reflect on the performance of their comrades in arms. When asked what portion of their fellow soldiers fired during any given engagement, the veterans estimated that about 84 percent of a unit’s men armed with individual weapons (rifles, pistols, grenade launchers, shotguns) and approximately 90 percent of those manning crew-served weapons (generally the M-60 machine gun) did so.

All the above from an article on Historynet.com

So, considering this war was fought by a famously non-volunteer force, this may represent one of the larger numbers of 'reluctant' soldiers. This article does not directly address firing without intent to hit, but I think the gist of the answer you're looking for is here.

Did some soldiers not fire, or miss on purpose? Definitely and possibly.

Did most? No.

As congusbongus wrote, this idea seems to originate in S.L.A. Marshal's Men Against Fire. I examined the book carefully a few years ago, and found that the whole idea seems to start in one scenario, which he generalised from WWII interviews with US soldiers in Europe.

An inexperienced infantry company is given orders to advance, without any kind of fire support. It does so, until it comes under effective fire from a well-concealed enemy, at which point the troops instinctively go to ground. At this point, nobody takes charge. The NCOs do not give the men orders to position themselves better, or organise fire. The officers don't give the NCOs orders. Nobody tells the commander about casualties, or provides him with any information.

Under these circumstances, 25% of the men firing in a useful way is actually pretty good. Marshall explicitly says that he doesn't think NCOs should be organising the men, but should be setting an example by firing. This doesn't work well when a unit is pinned down by enemy fire, because the men are at risk of being shot if they look around very much, so they won't see anyone setting an example.

Other armies don't expect as much individual heroism as Marshal seems to. When the unit goes to ground, there won't be much immediate fire in response. But the squad NCOs will be getting their men organised and passing information up the chain of command. Within a minute or so, most of the men will be firing, and they'll be doing it at selected targets, rather than the most obvious thing they could see from the positions they ended up in when they went to ground.

Marshal seems to have misinterpreted the problems faced by the US Army in WWII, where drastic expansion meant there was a terrible shortage of NCOs with combat experience. Interpreting the resulting difficulties as being due to fundamentals of human nature may have been very agreeable to the heads of the army, but it doesn't seem to have been accurate.

This idea originated in S.L.A. Marshall's controversial book Men Against Fire: The Problem of Battle Command, which claims that during WWII, 75% of US troops in actual combat did not fire for the purpose of killing. His ideas found a ready audience in many circles, especially the military.

But other scholars have since tried similar studies that reached the opposite conclusion: that most troops fired their weapons during combat. Other writers have cast doubt on Marshall's methodology, claiming that he cherry-picked or outright fabricated his data. There are personal anecdotes that paint him as an unreliable narrator.

But I have not found conclusive evidence to the contrary either. Also note that Marshall looked at WWII, which may be different to Vietnam (where many of Marshall's suggestions were already adopted), and different to modern wars involving all-volunteer forces. Unfortunately there doesn't seem to be rigourous studies in this area of military history. Marshall's numbers may be overblown but perhaps they contain a kernel of truth.