I would think a .22 WMR would take down a Coyote with a good shot placement and a hollow point would do even more damage.

In response to HuntingkidPA:
Here are some of the Rifle we are using right now here in Missouri to take out Coyotes and other Predators aka Bobcats,Foxes and Ground Hogs too.

All the following are Centerfire Rifles:

.22-250
.243
.223 (which is a very good Rifle,light weight,lots of different bullet weights/grains to choose from and it shoots very far out)
.220 Swift
.204 Ruger
.22 Hornet
.17 Remington
.17 Remington Fireball

I know a few Guys that have used their .270 or 30-06 Rifles too...a little too much for Predators especially if You wanting to save the pelts/furs and not do a lot of damage?