Don't have any coyotes here, so take this FWIW.

What about something in .223 or .243? Ammo is usually easy to get, shoots flat, minimal recoil, lots of models and platforms to choose from especially if you go .223.

.223 if you are planning on keeping the pelts, .243 if you don't.

Carbine style if you are up close, bolt action if they are further out.

Oh, and shotgun for really close!