Will a 22 LR kill a coyote? Yes.
Is it ideal? Hardly.
Would I use one to hunt coyotes? Only under narrow circumstances. If game laws restricted carrying a centerfire during deer or elk season (like they do here if you don't have a deer or elk tag or used them up), I'd rather use a shotgun instead. With the right shell and choke, you can take coyotes at 40 to 50 yards, and much more effectively than a 22 LR.
30-06 with light bullets will work fine. At close range, though, you'll probably blow the coyote in two. Pelts won't be worth skinning.