I know a lot of old-school hippy-crunchy Phoenicians on strict veggie and vegan diets, and they do not have trouble eating out. IMO, the Tempe area is a little easier to feed veggies in than elsewhere, but I've never had actual difficulty finding a restaurant with awesome vegetarian options.
Pita Jungle has several restaurants throughout the area. They offer an substantial vegetarian and vegan menu. www.pitajungle.com
I haven't been there in a while, but the Supreme Master Ching Hai Vegetarian House was always a great place to go eat with vegans (3239 E. Indian School Road).
Cafe Lalibela in Tempe is Ethiopian and has tasty, tasty vegetarian options.
azcentral.com has a dining search feature. Running a quick search on "vegetarian" yielded 191 returns.