The only fully vegan eateries in K-Town are Sanctuary Vegan Cafe and the VietGrill vegan 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).
Local Restaurants & Local/Regional Chains
Bida Saigon (now offering veggie broth for pho)
FIVE Knoxville (they've removed some items from their menu, but it still says "vegan options available," so....)
Mediterranean Delight (Oak Ridge)
Southern Sisters Mobile Cafe (food truck)
Three Rivers Market (hot bar and salad bar)
VietGrill (all-vegan food truck)