The civilian version of the AR-15 is a poor rifle to use for deer hunting. It’s not deadly enough. If you are hunting, you want something that can reliably kill a deer, not just wound it. Contrary to popular belief, the AR-15 is not a super-deadly high power military rifle. The reason people don’t use it for hunting is that it’s really not a very good hunting rifle. Some states actually don’t allow the 5.56 cartridge to be used for deer. Other states do allow it, but it’s on the low end of what is acceptable. If you do use it for deer, you really need to be within 100 yards of the deer. Anything longer than that and the bullet will have shed too much energy to reliably kill the deer.
The fully automatic version of the AR-15, which was adopted by the military as the M-16, uses the 5.56 NATO cartridge, This cartridge is an “intermediate” cartridge. It’s not as powerful as a rifle cartridge, but it’s more powerful than a handgun cartridge. The military uses intermediate cartridges in assault rifles because full size rifle cartridges have too much recoil and are difficult to control in full auto mode (lesson learned from the M-14). Logistics are easier with smaller and lighter rounds. For the same size and weight, cargo planes, ammo trucks, and even the soldiers themselves can carry significantly more rounds of the smaller intermediate cartridge.
The civilian AR-15 is designed to be the civilian version (and therefore the non-assault weapon version) of the military M-16 (which is an assault rifle). You can use it to hunt smaller game, to protect livestock, or for self defense. It is not designed to hunt larger animals like deer.
If you really like the design of the AR-15 and want to hunt deer, you are much better off getting an AR-10, which is the predecessor of the AR-15. The original AR-10 was also capable of full-auto, but used the higher power 7.62 NATO cartridge (which is basically the same as the .308 cartridge). The AR-10 was never adopted by the military (it evolved into the AR-15 instead, which was adopted) but you can currently buy civilian versions of the AR-10. The civilian versions are of course semi-auto only.
I looked up the most popular hunting rifles and found a list of the 25 most popular hunting rifles of the last 50 years. The Ruger Mini-14 was the only semi-auto on the list.
The Mini-14 is basically a scaled down version of the military M-14. It is semi-auto only and fires the 7.62 NATO / .308 round (with some other cartridges available) and is often used for hunting as well as protecting livestock. Designed specifically for civilian use, the Mini-14 is actually a much more deadly weapon than the civilian AR-15.
There were two lever-action rifles on the list, and a modern black powder muzzle-loader. The muzzle-loader might surprise some folks, but there are hunting seasons specifically for black powder muzzle-loaders, and most hunters who hunt during these seasons prefer modern weapons over old fashioned Civil War style muskets.
The rest of the rifles on the list were all bolt-action.