I hunted varmints with 22 long rifle a bunch of times when I was young and never had good luck with killing power - other than a few head shots.
I don't recommend it unless it's the 10/22 Mag. I love using my 22 Hornet for small game, but it's about as small as I feel comfortable using to make sure I get a good amount of energy and cavitation in there. Coyotes are a little big for the 22 long rifle.
If you do use a LR for song dogs, then here's what I suggest...
1) Close range only - inside of 50 yards
2) Use CCI Stinger ammo
3) Be ready for successive, accurate followup shots (goal being a quick kill)