I have hunted far more small game animals in my life than anything else, and I can say that I have probably killed hundreds of squirrels, dozens of grouse and more than a few other animals of various sizes and most were shot with a .22 and mostly with standard velocity ammo because of the noise and I used to buy it by the brick because it was cheaper and easier to get because while it worked great in my single shot Remington most semi auto firearms won't cycle it.
I try for head shots but that is not always the case but a head shot well placed is lights out. I have hit many animals in the vitals only to have to occasionally shoot them again but not always. Last year I spotted a large woodchuck on my property and in that case I used a CCI Mini Mag .22 and got him in the head at about 40 yards, he rolled over and was dead in seconds. Squirrels, well, if the shot is clear, I will hit them in the vitals, most fall right over, I would say 90% when it's a good hit and again that's with standard velocity ammo.
One animal that I was really surprised that I did so well on was a cottontail rabbit I shot a couple years back at about 40 yards. I hit him in the right side behind the shoulder, again, standard velocity .22 LR. He rolled over dead and went no where. What was really surprising was the exit hole, again this was not a high velocity round, it made a large hole in his left side as the bullet exited, you would have thought he was hit with a .22 Magnum or maybe even a .17 HMR because the bullet hit no bones on the way through.
Your best bet is to aim for the head if you want a hit that is going to be fatal almost always. Woodchucks can be tough sometimes and can take a couple .22's to the body but a well placed head shot will put their lights out. For them though and animals that size, I would use good .22 hunting ammo like a CCI Mini Mag or something that has similar performance.
The last woodchuck, got it with my old Remington 33 single shot and a CCI Mini Mag to the head. Never took a step.
This was the rabbit I got with the same rifle using standard velocity ammo.
This is the same rabbit and this is the exit hole. Again, this was just a .22 LR standard velocity round, nothing going that fast. The rabbit was dead instantly, but this was more or less a fluke. If you want your animal to have an instant lights out, put the round in the head if you can. I never had one of those crawl away yet.