A belt-fed shotgun is totally possible with the proper R&D. Bigger and more technical problems have been solved than feeding 12-gauge shells through a belt-feed mechanism. I know of a few transferable belt-fed MG shotguns out there, just some C2 tinkerers from the 80s who wanted to play with the concept.
I think the fatal flaw is in the application. This would be a fun toy, no doubt, but that would be about it. It wouldn't have any military or law enforcement application. Think about what a belt-fed weapon is supposed to do and you'll realize that the shotgun doesn't fit in those paramaters. Hell, a magazine fed shotgun is a better idea and that never took off despite numerous attempts. The USAS-12 is the perfect shotgun in this regard. It's reliable and heavy to mitigate recoil, it feeds from a magazine so reloading is easy and quick, and it allows using multiple types of loads far better than anything else. But it still never took off.
No one would fund the serious R&D when the sales prospects are so limited.
Another fatal flaw in a belt-fed shotgun would be the ammunition itself. While shotguns generate less heat than just about any other firearm, they do generate heat and a plastic hulled shotshell will melt inside the chamber and barrel when it gets hot enough and this will shut the gun down for a long time.