There has been a lot of discussion about this, mostly inspired by Dave Grossman's On Killing: The Psychological Cost of Learning to Kill in war and Society and a number of other books such as Melanie Joy's Unnatural Born Killers.

Opinion on the validity of these claims is divided to say the least and hence there is no clear answer. My personal opinion based on the research I've done into developing military tactics tends to be that while there may be some truth to the claims there was not a systemic issue with soldier's deliberately missing. Instead, warfare changed so that the objective of pulling a trigger was no longer primarily to kill the enemy, but rather to keep their head down.

The advent of cover-based conflict was still fairly young at the dawn of the 20th century. Previously firefights would take place at much closer ranges and so the effectiveness of firearms was much greater. However with the advent of more and more accurate and quick firing infantry weapons and machine guns in particular it became incumbent on soldiers to take cover and prioritize self-preservation over annihilation of the enemy.

Having a target in the open was a massive rarity during these conflicts and the majority of firing was done to prevent the enemy from maneuvering to an advantageous position where they could overcome your cover. In WW1 in particular when troops went "over the top" and left cover, the slaughter was horrific and opposing soldiers most certainly shot to kill. Whole platoons and even companies were wiped out in minutes, hardly what you'd expect if there was a systemic determination to miss.

Ask any modern soldier how often they have a target in the open and they'll tell you it's a tremendous rarity.

Instead of firing at the enemy, you fire at their cover hoping to pin them down and gain a maneuverability and encircling advantage or, these days, hold them in place long enough for a Warthog to blitz their position. In this way, yes, I suppose the soldiers do not fire to kill, but they're certainly not trying to spare the opponent, it's just that firing to kill is no longer directly possible the majority of the time.

People point to statistics like the 50,000 bullets fired in the Vietnam War for every combatant killed as evidence to support this claim but fail to understand that the majority of combat under these circumstances involved concealed soldiers firing at each other with no real idea of exactly where the enemy was. You fired at the building you believed them to be in hoping to saturate it with enough bullets that one of them would get lucky and kill the bugger.

Again, it's one of those questions that's pretty much impossible to answer with certainty one way or the other as it asks us to speak to the psychology of people in events almost a century ago. But I've never seen any really compelling evidence that it was a systemic issue.

Edit: I would highly recommend that everyone who reads this also read /u/Georgy_K_Zhukov 's excellent response below which goes into detail about many of the methodological and research problems that On Killing faces.

I posted this as a response originally, but with a bit of edits, it does address your question itself, so reposting it as a top level

Over the past decade, this claim has become quite prevalent due to the work of Lt. Col Dave Grossman, in his book On Killing. It is a interesting book, certainly, but incredibly controversial. His conclusions come from a synthesis of other sources, and the central argument is that firing rates are very low and this is because most people have a very compelling urge not to kill which must be overcome. This isn't really his own work though, and the core of his argument comes from Men Against Fire, a work by S.L.A. Marshall, a US Army historian. Marshall claimed that, after conducting extensive interviews with US Army units, often immediately after they had been in conduct, he had discovered that only 15 to 25 percent of men actually fired their weapons with the intention of killing, and the rest either missed purposefully or just didn't fire. This statistic is really the heart of Grossman's book in many regards, and he takes it to be the word of god, for all intents and purposes.

Marshall's study did have a major impact on training by the US Army, but in the years following its publication, pretty much everything about it has been called into question. There is no end to the pieces you can find published in the past few decades which essentially rip him to shreds, going from just questioning his methodology to outright claiming that many of the interviews he gave were essentially fabricated. [A few] (http://strategicstudiesinstitute.army.mil/pubs/parameters/Articles/03autumn/chambers.pdf) quick examples I dug up. Roger Spiller's "S.L.A. Marshall and the Ratio of Fire" seems to have been the first full assault on Marshall, but I can't find a copy online, just this which has some quotes.

Grossman draws on other sources as well - mentioning French studies done in the 19th century, the recovery of unfired weapons at Gettysburg, studies done by the noted British wargame designer Paddy Griffith, and most interestingly, supposed laser reenactments of historical clashes done by the British Army in the 1980s which confirmed low firing rates (I have searched high and low for these by the way, cause it sounds awesome, but never with the least bit of luck. If anyone knows what the heck Grossman is talking about, please let me know!). So anyways, Marshall isn't his only sources, but it is the course of his argument at points, and you can find plenty of sites, both professional and amateur, taking Grossman to task for relying on the numbers so uncritically.

And really, that's perhaps the most important part. Using the numbers isn't a sin in of itself, it is passing them on without the least bit of mention of their controversy. It points to Grossman finding evidence to support his preconceived notions rather than having evidence point him to a conclusion. He cherry picks and ignores what he doesn't like. He also doesn't entertain similar objections that can be made about other evidence he uses (see /u/BOMPendragon about the Gettysburg arms). Especially in the case of Marshall, his study was essentially dropped so long ago, so not mentioning the controversy is a real disservice, as people won't even be aware of it anymore!

Marshall's work does have its merits, and even Grossman's does, as I find his book very interesting (Avoid On Combat though). He does use first person accounts to good effect (he relies on Keegan a good deal IIRC) to demonstrate his ideas, and there really is no lack of literature on the topic that points to some hesitation on the part of some soldiers to engage in combat, and the internal demons that must be wrestled with. Grossman just does a terribly inept job of investigating that, and lets his own biases get in the way of a proper study. As for Marshall, I read his book ages ago, and he too brings up important points that have merit, but again, we can't overlook the shoddy job he did on some of his central points.

So to directly answer your question, there is a lot of literature out there that makes the argument, and a lot which questions the former group, the former being popular in the post-WWII era and the later a reaction beginning in the 1980s. I'm not going to get into my opinion on the matter exactly, since I do have one, but simply say that it isn't a settled matter of fact, and I doubt you will see the argument put to rest any time soon (although opinion has certainly been shifting away from Marshall, Grossman's resurrection of him not withstanding). The best you can do is read the opposing arguments and evaluate their merits.

Edit: After a bit of digging, here is Grossman's response to the criticisms of his use of Marshall.

Some reading for you. SLA Marshall and the Ratio of Fire.

Killing for Their Country; a New Look at Killology

TL;DR

SLA Marshall's research was not enough to prove anything. Documents about Canadian infantry at the same time showed that inexperienced and frightened soldiers were apt to fire too much at the enemy, wasting valuable ammunition as their survival instinct kicked in and they did nearly the only thing they could to survive -- shot back.