"I would like to make coyote pelts."
Folks haven't been reading that, apparently.
For handloaders, a .223 with 35- or 40-grain bullets likely would not have exit wounds on a body shot.
Otherwise, I know one hide-hunter who calls them in and uses a .22 rimfire magnum--but he tries to hold his shots to around 50 yards.
Headshots with a .22 rimfire should work okay.
You're from BATFE? Come right in! I use all your fine products!