If close, 300blk, if further, .223 or even .243. .243 is ballistically a fantastic round, and would have plenty of energy for a coyote even 600+ yards down range.

For a semi-auto, Maybe a Remington Woodsmaster in .243. The new versions are known to have reliable cycling, and between the gas system and the new model recoil pads they are using it should be pretty soft shooting for sure.

A bolt will kick a bit more, but cost less. Can't go wrong with a trusty Rem 700. Between being slightly lighter and not having a gas system to pull any recoil, it won't be quite as soft. But any of the above calibers with a proper recoil pad will be pleasant to shoot.