as has been said a dozen or more times already, yes the little lowly .22LR can take deer and other animals. Is it recommended or even considered effective? Nope. Not by a long shot. It IS, however, far more effective than sharp sticks, big rocks and a whole bunch of other improvised weapons.
The .22LR has a place in every survival kit. It has several really very useful characteristics:
1) It is cheap. A really good .22 rifle is a mere $200. I've purchased a new Mossberg .22 semi on a black Friday sale for a mere $64 last year. There is no reason for not having one.
2) The ammo is small, light and inexpensive. It can be stockpiled in quantity.
3) The rifles are light, easy to handle, have little noise and low recoil. They can be used by just about everyone.
4) They are highly effective on all manner of small game. Rabbits, squirrels, groundhogs, porcupines, racoons, grouse, turkeys, geese, ducks, muskrats, beavers, whatever... If its under 50 lbs it's in serious trouble.
5) While the .22 is likely the least effective round we have for big game (excpet maybe for .22 short, .25 acp, and some of the .17 rim fires) it can be used with moderate success and it is definitely better than no gun at all....
6) practice is fun and cheap with .22's.
If you live in a major urban center (meaning there is little to no game except rats
) you are likely best served with a defensive firearm first. However, if you live in a rural area, I'd argue that a good quality .22 semi rifle (Ruger 10/22) should be the first survival firearm you buy.