In a town where any breakfast, but especially vegan, is near impossible to find before 9am, Cup Cafe is a real gem and open at 7am. Located in the historic Hotel Congress downtown, the atmosphere is as great as the food. There is a single vegan option for breakfast (Tofu Scramble, $9.5), but its amazing and the best I've had in Tucson. They make their own vegan sausage in house and the very knowledgeable staff knows which side dishes and bread are or can be vegan. They also have soy milk for your coffee.
There are additional options for lunch, dinner, and even on the cheap bar menu, but I've only had the Queer Steer ($13 for dinner), which is also delicious. In the midst of all the new pretentious and meat-heavy, farm-to-table restaurants downtown, it's great to have delicious options at the coolest old bar/hotel/restaurant in Tucson.
Cons: parking - its downtown, a little pricey