Hiking the necessary miles each day to finish the AT in one hiking season takes a lot time each day. Hunting and foraging take lots of time. There are not enough hours in the day to do both, especially if you are burning up to 5000 calories a day just by hiking 20+ miles with lots of elevation gain and loss.

It takes bales of vegetation to get 5000 calories - just ask a giraffe! Hunting is subject to restricted seasons for most game species and there are many local regulations. Hunting is not allowed at all in National Parks and foraging is probably frowned on, too, and may be entirely forbidden (I'm not sure on that - you'd have to look it up).

These are just a few of the reasons you will not find anyone who has tackled the AT by relying entirely on the surrounding wildlife and vegetation for nutrition. I'm sure there are more, but these few are enough by themselves.