I agree with xxkid123 and Bill that it's mostly an accuracy problem.
Hunter talk, if you're squeamish don't read:
I don't doubt that a sling stone will break bones, even deer bones. What I doubt is that most people can catch a white tail deer after breaking one of its bones, or even puncturing a lung, without a pretty substantial blood trail to follow (which sling bullets don't do as well as broadhead arrows or bullets do). Deer bolt, fast. Even when they shouldn't still be standing and are about to die, deer can move faster than the average person IMHO. You'd probably have to hit the deer squarely in the eye, the gut away from the ribs (which would ruin a fair bit of meat if the intestines burst), or in the jugular in its neck (iffy) to kill the deer fast enough to have a chance at catching it and I think those shots are too difficult to expect to pull off. Odds are more likely that you sling at and hit the deer in the ribs and puncture a lung, it runs a hundred meters or so (in dense forest), it dies, and you spend the night searching for a deer carcass that may or may not ever be found. Same thing that happens when you shoot at deer with a .22, and definitely not worth the effort if you have any other choices available. I'd definitely focus on trying to find some other animal, like squirrel, turkey, or maybe even javelina (if I'm
really desperate or have a death wish) which aren't as resilient as deer. And in the case of javelina, they don't spook as easily as deer do. You may have to learn to sling at something
charging at you, but that just makes it more interesting.