Another visitor to Taipei
If you'd like to try Chinese-vegetarian, Yuan Shu is pretty good. But I was last there more than 3 years ago - a German colleague was vegetarian, so our local Taipei colleagues brought us to this place.
Maybe get your hotel concierge to call up the number below and check on the restaurant first before you make the trip, just in case.
P.S. - Taiwanese have a large Buddhist population, many of whom have opted for a vegetarian diet, so you'd normally find meatless sections on Chinese restaurant menus, and getting vegetarian food in Taipei is much easier than in many other countries.
Yuan Shu Vegetarian
2, Lane 14, Lishui Street
(Address for taxi driver: 大安區麗水街14巷2號)
MRT station: Guting
Vegetarian is easy in Taipei, once yo've figured out how to explain it. At the cheap end, you've got Buddhist buffets, which have a lot of variety and can be dirt cheap (lunch for under $2).
Is a higher end vegetarian buffet I've taken work visitors to - mainly vegan, a bit of milk based stuff, on the organic food line. Good selection, 600 NT (about $18) for an adult for dinner, including non alcoholic beverages and desserts. It would be a short cab ride from the 101 area.