To find a restaurant near where you are right now, check out the map (note: not everything here appears on the map, due to incomplete Google listings).
The only fully vegan options in K-Town are Sanctuary Vegan Cafe and the Dinner Bell Fresh food truck. That said, there are lots of places that offer one or more good vegan options. Some of the restaurants below offer lots of choices, while other have just one or two things. Some may call for simple changes like removing cheese or mayo, but I haven't included any that require major modifications (like taking the chicken off of a chicken salad).
If I've missed one of your favorites, please shoot me an email!
Local Restaurants & Local/Regional Chains
Bida Saigon (now offering veggie broth for pho)
Dinner Bell Fresh (food truck)
FIVE Knoxville (they've removed some items from their menu, but it still says "vegan options available," so....)
Mediterranean Delight (Oak Ridge)
Three Rivers Market (hot bar and salad bar)
Viet Grill (food truck)