I've had a Wrist Rocket slingshot (the original wrist-brace slingshot) for just about ever. I shot them as a kid, and lost them, and then replaced them. I NEVER got any good with them, though.
Still, I thought they would be neat for a survival kit. The only problem is that Wrist Rockets, and any other kind of braced slingshot (even the folding kind like I have) are just too big ot fit in a pouch easily. So, I took a hacksaw to the folding wrist brace and removed it. This allowed me to shoot the slingshot in a proper upright manner, which the brace hadn't allowed; I was always shooting the thing cocked over to the side, and was unable to get a consistent sight picture. To replace the brace, I used a piece of 550 cord, tied like a lanyard in the hole where the brace had been. My slingshot is now MUCH more accurate and easier to shoot, and will also fit into a small pouch easily.
As for ammo, I've been using .22 and .30 ball bearings latley, and was shooting against a target backstop made of a carboard box stuffed with old magazines and covered in duct tape; very tough target that I use to stop field points from my bow. The .22 and .30 ball bearings punch into the box, no problem. I don't know how far they're penetrating, but I would guess that they would shoot through small game (rabbits and birds)pretty easily. Marbles are by far the cheapest ammo, and give good results, although you're looking at blunt-force trauma, since a marble won't penetrate. But a marble in the head should put a rabbit down nicely. I haven't tried the 3/8 (about .40) ball bearings, due to cost, but I may get some to put in my kit. They sound like a good "compromise ammo", as has been pointed out.
The only downside of using a slingshot in a kit is that the bands rot awfully fast. I can;t seem to get a pair of surgical tubing bands to last over a year or so. Does anybody have any "tricks" to keep their bands from rotting?