shotgun
I shotgun 'yotes some, usually by accident turkey hunting. (Coyotes accident, gobblers good fortune)
But, if I'm calling coyotes in the thick stuff on purpose, I use 2-3/4 " - #4Buck. Just happes I've got a supply of the stuff and it works. I've bought some of the Dead Coyote T shot, but am reluctant to shoot it through my vintage and beloved Wingmaster and its turkey choke.
If I were to buy a dedicated varmint shotgun (unlikely) I'd look a hevishot /T shot and the correct choke tube for same. I ever see a deal and a half on a Mossberg, it might become a varmint shotgun, but again, not likely.