the first coyote I shot, back in 78 or 79, was with a 22lr. I was actually rabbit hunting around our farm when he came running by. It was a snow day from school, so I was hunting. BUT anyways, I shot 3 times at the coyote as he went by at about 50 yards. It never once showed any signs of being hit, but I got on it's tracks to follow it, hoping to get more shots at it. I had tracked it a couple hundred yards when I came upon it....dead. Not one drop of blood in the snow to show it was even hit. I'd guess I hit it in the lungs.
Moral of story, the 22lr will kill a coyote, BUT effectively, NO! Unless of course the coyote is in a trap and you're using the 22lr to kill it! I find most guys can't center punch the head most times, so even to say "head shots only" isn't practical on called coyotes!
Fox and bobcat arn't as tough, BUT even on them, I'd suggest something with a little more punch!