I keep meaning to pick up a slingshot from this guy: A+ Slingshots. He also sells lead ammo that is intended for hunting.

However, as others have said, I think a lot of practice would be required to consider it a viable survival tool. That said, it is relatively light & compact which in my mind makes which makes it a great choice for this if you can pair it with skill.

It depends on where you live too. In discussions like this someone often points out that a .22 pistol is a better choice for the weight & size and they'd be 100% correct. However, for those of us in Canada, that simply isn't an option. Heck, in a lot of the wilderness areas I frequent, firearms of any type are prohibited. So, for us, I think the slingshot has definite merit as a survival tool.

That said, if I were to get serious about wanting to catch small game in a survival situation, I'd want to also develop my skill with various trapping techniques.
_________________________
Victory awaits him who has everything in order — luck, people call it. Defeat is certain for him who has neglected to take the necessary precautions in time; this is called bad luck. Roald Amundsen