I've killed a ridiculous number of deer using a 223 with 55 grain soft point ammo. I've shot some with I believe 62 grain bullets as well. Shoot them in the lungs toward the lower third of the chest, they'll bleed.
62gr federal fusion is heck on a deer, all my kids shot for several years. Never lost a one, never had one go past maybe 75 yards. Now I am smart enough to know when they were big enough to shoot something bigger, to shoot something bigger, but the ar platform worked well for us when starting out.
Barnes VOR-TX Ammunition 223 Remington 55 Grain TSX(Factory loads). I have killed a lot of deer with BarnesTSX 53gr with my .222 including heart/lung shots as well as neck. Also you get complete pass thru with a nice exit. My son shot his first behind the shoulder at 8 years old. It ran 30 yards and piled up.
I've been trying to come up with a solution for my two youngest to hunt deer. They are 6 and 7 and both small of stature and struggle with good form on even the shortest youth models. On a hunch, I collapsed the stock on an AR and low and behold, they both shoot it comfortably and very well with a red dot.
Problem is, as you all know, there are virtually no .223 rounds to be found anywhere in America. I've got an assortment of rounds ranging from 55 up to 62gr, none of which I feel confident shooting a deer with. However, I have a few boxes of 68gr Hornady BTHP that I can load up. Hornady doesn't recommend them for hunting, but they're the closest thing I've got in .223 to a lethal deer load.
I found loaded rounds for .260 & .270 that I have been looking for for over a year...I get the can I find them or not . One was 110TTSX in the .270 and 120TTSX for the .260 along with about 75 rounds of my LR 142gr .260
My daughter hunts with my Ruger Precision in .308. Between weight of the rifle and muzzle break, recoil is manageable and she's small. Point being, if you can't find bullets/ammo for the rifle you intend for them to use, maybe look at adding a mod to a different rifle?
These are the Sierra 65 grain game king bullets that I reload and have used for deer and they drop in their tracks when hit in the lungs just above the heart. This is a picture of my M4 and a 5 shot group I shot at 100 yds. with a rest.