No, goats aren't allowed on the AT.
Goat hiking is way too cool. I've seen a few people out here with goats. There are places in Estes Park that rent some kind of pack animal, but I can't remember what. You take a half day class to learn how to pack them and take care of them, then you go hiking with them. I posted about it here a while back, just can't find it. Still goats are better cause you don't have to bring food for them, they just graze from what I understand.