How difficult is it just on a practical level? And do you have to eat mutton? I'm a life long vegetarian
Practically it's pretty easy - everything's connected by a surprisingly effective system of semi-scheduled minibuses called marshrutkas, and there's enough tourism, at least in the places that we looked at going, that you can find places to stay and to eat without much trouble. Picking up very basic tourist Russian helps a lot.
My girlfriend is veggie and got along okay - there's a lot of meat in the food, but stuff's normally prepared on the spot and restaurants seem to have no problems leaving it out. Might be a bit harder if you do a lot of homestays or if you're scrupulous about checking what sort of stock they've used or whatever.