I have shot groundhogs with 22LR in the past. You pretty much need to limit the distance and always shoot for the head (not heart-lung area). The 22 drops them fast with head shots, but the whistle pig usually runs into its hole with body shots to die later. I always tried to stalk or choose shooting locations that allowed me to be within 75 yds or a distance where I am pretty capable with head shots.
The 22WMR would be better or perhaps the 17 HMR. I hesitate to recommend the 17 due to the bullet weight.
The newly re-introduced 5mm would be the best choice of all if choosing a rimfire. The higher velocity than the 22WMR and the heavier bullet should make it more effective than the 17HMR. I'm going to buy one when the new rifles start hitting the gun shops. I figure it should be good chuck medicine out to 100 yds or so. But I will have to experiment a bit on distance and bullet impact to the animal.
All told, you are better choosing a 204, 222, 223, 22-250 kind of centerfire caliber for chucks at reasonable distances. I'd go with the 223 myself.