I own a Colt AR-15 Sporter chambered in the unusual caliber (for an AR) of 7.62X39mm. I use it for hunting deer in wooded terrain. Because of the taper of the cartridge case, a magazine that would normally hold 20 rounds of 5.56mm NATO ammunition will only hold 5 rounds of 7.62x39mm. Other than being semi-automatic and having lower ammunition capacity, it’s the functional equivalent of a typical .30/30 lever action rifle.
When I first moved to Maryland in the early ’90s, I had hunting privileges in an overgrown orchard next to a 6,000 acre no-hunting environmental area and a 200 acre corn/soybean field. The woods were dense enough that the longest shot possible was less than 100 yards, so a .270 Win. or .30/06 was overkill. I used a .45/70 rifle for the first few years but decided that something like a .30/30 would make more sense. I went to a gun show looking for a deal on a Marlin 336 lever action rifle and stumbled on a dealer who had a Colt AR-15 in 7.62×39 mm that he wanted to sell cheap. The price was right, and I bought it. It’s been a nice woods gun, performing well with Winchester 123 gr. soft points. I’ve had even better results using the now-out-of-print Hornady Zombiemax and the Hornady 123 gr. Black ammo.
So, yes, you can hunt deer with an AR-15. I’ve been using mine for woodland hunting for almost 30 years.