If the law states .22 caliber or smaller then you could use any of the following:

rimfire:
.17 Mach2
.17 HMR
.22LR
.22 Mag

centerfire:
.17 Rem
.204 Ruger
.220 Swift
.221 Fireball
.222 Rem. Mag
.223 Rem
.22-250 Rem

I don't think that I would try to use a pistol to hunt coyotes myself, but I am sure it is done. I have killed many with a .22LR, but if you want to be able to shoot at longer ranges you would be better off with any of the centerfire cartridges.