Do you have the TIL link for sourcing? We'll know a lot more about the veracity of the claim that way.

OK, so there seems to be precious little information out there on this. A few other users have done an excellent job tracking the citations back, and jumping into the fray myself I was able to find some discussion of the claim in "US Combat Shotguns" by Leroy Thompson, who focuses on two sources, Milton F. Perry and Paul Jenkins, who both wrote articles on the use of 'Trench Guns' during World War I, and both who make mention of the claim, which we can say with confidence dates back to the war itself, as an advertisement for DuPont from 1918 makes mention of the practice. He quote extensively from both. Perry, writing in 1956 (Found the article), recounts:

With the utmost secrecy, trap shooters were recruited, trained in a hush-hush program, and rushed overseas. At Chateaux-Thierry [sic] modern shotguns red with world-wide reverberations, though the public did not know it until afterwards. A German attack was imminent; and it was the drive of 1918. Americans were rushed to bolster the line: marines, infantry and shotgunners.

In outposts ahead of the main line of resistance we placed some of our very best trapshooters – men who could hit a flying bird upon instantaneous reaction. In the trenches was a unit composed of those who had shot at sporting meets back home. Breathlessly they awaited the assault.

Their first warnings were German “potato masher” hand grenades lobbing through the air. Few landed as most of them were exploded in the air by the experts in the outposts. Upon the failure of the grenade attack, the enemy launched a mortar attack. Again the trapshooters proved their worth, de ecting the slowly arching bombs. Finally, a vast grey [sic] wave of the Kaiser’s best surged forward.

However, Thompson doesn't find this to be too credible, or at least the claims about it being so organized, and finds other claims made by Perry, such as a claim about 50 rounds per minute, to also strain credibility. While he doesn't discount the feasibility of using a Shotgun to take out incoming grenades, he does question how consistent such a program could be, and furthermore that given the lack of records, it seems that at best we can say it was not used often.

As for Jenkins, writing earlier in 1935 (My efforts to find the article, as others, also failed), he sought out to evaluate the claim, and at least found it to be technically feasible, but has his doubts about the practice:

We found the [training] bombs could not only be hit, but we shattered one of those cast-iron dummies in the air, with buckshot? Yet let me rise to remark that, while it is alright to stand and shoot at the incomers at skeet stations 1 and 7 – IF the said incomer were an iron bomb weighing 27 ounces and timed to explode in 5 seconds from its start, and you had 20-inch cylinder-bore shotgun with 9 pellets of shot with which to hit it and knock it away – what would you do? I cannot escape a strong conviction that all world’s sprint records would be broken in the next instant of time by somebody moving, so to speak, away from there!

Thompson apparently attempted to replicate the experiment, and had almost no success deflecting the grenades, but as they were duds, we can't discount the fact they might have exploded even if not deflected.

So at least as far as Thompson goes, the sum of the argument is that there are several mentions, dating back to the war itself, of shotguns being fired at incoming grenades, so it is quite likely that it did happen, if only in a few isolated attempts. The claim about professionals and organization seems to be much less likely. While he may have simply missed a source, the first mention of this aspect that Thompson tracked down comes from the 1950s. If anyone has success getting the entirety of the Perry article, or of other early sources repeating the claim, it might help clarify things, but right now the best we can say is "partially true".

Edit: Trawling through old issues of American Rifleman to hopefully find mention, no luck yet. I did, however, find in Vol. 64 No. 14 mention that trap-shooting was used as part of the training program for Airmen, as it helped train them about shooting at moving targets. Of additional interest, while the article notes approvingly that the shotgun is a popular weapon in the trenches, and that there are "100,00 trapshooters in the National Army", it doesn't make mention of this particular usage. But of course, in March 1918, when the issue came out, Americans had seen minimal combat.

Edit II: SUCCESS! Sept. 29, 1917 Vol. 63, No. 18 of American Rifleman (Then still called Arms and the Man), page 358. Emphasis mine:

Club Teaches Public to Shoot"

The Nebraska State Sportsman's Association is thoroughly alive to the needs of the times. It is a most energetic body. Some time back it elected its president to the Governorship of the State, and now its secretary appeals to every club member in a bulletin to take up trapshooting as an aid to the Government.

The bulletin, which is appended, is of interest to every gun club and State Sportmans' Association, and it would be a good move if many other State associations followed the example of Nebraska:

“Trapshooting formerly known as the ‘Sport Alluring’ has willingly had its name changed to the ‘Patriotic Sport.’ Why the Patriotic Sport? Because the United States Government has recognized it as a long step toward preparedness and has adopted the shotgun as a most valuable defense weapon. "Think of the chances of the enemy getting into our trenches for a hand-to-hand conflict with a bunch of American trapshooters stationed there with magazine shotguns, each containing 10 loads of buckshot. Shotguns are also to be used for the destruction of the handthrown grenades.

“Thousands of trapshooters have gone to the colors and when they arrived at the training camps they took to the army rifle like ducks to water. Officials in the army said if all recruits were as prepared as the trap shooters they could step to the front and win the war in a few months. The Government -recognizing this immediately established traps at every training camp. Their first order was for six million clay targets, some of which went to the boys in France. Expert shooters are at all camps and aviation schools teaching the boys to shoot.

"It is the patriotic duty of every club in our association to open their traps to the public with no charge except for actual expenses, and it is the patriotic duty of every member to get the boys who have registered for services to the traps and teach them to shot. It will not only save the Government the time to prepare them after they arrive at the train ing camps, but it may actually save their lives.

"I feel it is my patriotic duty as your secretary to insist upon prompt action by your club, and if you feel the thrill that I felt upon learning the Government had recognized our favorite sport as the Patriotic Sport, you will get busy. Do not let anyone tell you it is not right to shoot as the ammunition you are using may be needed for the army. Just the opposite, it is not conservation, but preparedness. Shotgun ammunition is going to be manufactured as a Government necessity."

So I would stress this dates to late 1917, before any Americans saw combat, so this doesn't speak to the actual employment in battle, but it clearly demonstrates that this was being discussed in theory at least even before the Americans entered the trenches.

While we certainly have incontrovertible evidence of explosives being used as means of transport during the First World War, the case for grenades being detonated mid-air by sharpshooters is clearly far less straight forward, judging from the large number of incorrect, speculative or otherwise disallowed comments in this thread. This means we've had to remove a lot of responses so far, including plenty asking about where all the other responses went. Please don't be That Guy.

Ladies and gentlemen, please keep in mind that this is /r/AskHistorians. Here, we have stringent standards which require expertise and depth when answering questions. Please refrain from making speculative, vapid, or other poor responses, or else you will find our retribution swift and brutal.

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^^^Sources: ^^^1 ^^^/u/TheToquesOfHazzard ^^^2 ^^^/u/Shizcake ^^^3 ^^^/u/1halfazn

However, Thompson doesn't find this to be too credible, or at least even before the Americans entered the trenches.