I suggest Mi-le Su Shi (Mi-le Vegetarian), in the National Taiwan University area. I don't know the exact address, but here's how to find it:
Take Green subway line to Gongguan station
Walk North on Roosevelt Rd., and turn right onto Shinsheng South Rd. You want to end up on the NE corner, which means you will need to take the underpass. Walk past Eslite Bookstore on the left, and keep walking until the next alley on your left. Take a left into that alley and look up, there should be a sign overhead. Head up to the 2nd Fl.
Mi-le is not a fancy place, but its food really fresh and good, with much variety to choose from. Self-serve at the buffet table, and bring your selection to the cashier, who will tell you how much it comes to. Oh, and here you ask for rice too -- white or mixed grains. Soups are free.
I was in Taipei for only 3 days....too short, and too last minute, but I did manage to walk by a vegetarian stand at one of the 4 corners at the Shuanglian Subway station. Though I didn't get to try their food, I saw that they had what i'd been looking for for over 24 years (that's how long it's been since I've been back in Taiwan), the Haw rounds that are totally different form the ones you get overseas. These are pasty pale in color, 1/8" thick rounds in a plastic bag that they sell for 30 NT. It's a bit sour and herby. Taste of childhood memory....
Anyhow, I would have tried the vegetarian food there if I hadn't been full with many other long-time-no-eat street eats. I have to say though, that it's rarer to have vegetarian street food available in Taiwan, so it may be worth a try.
In the area underground between Shuanglian station and the Taiepi Main Station (with Zhongshan Station in the middle) there is the long street of underground bookstore, and as it gets closer to the Taipei Main Station, all sorts of shopping to be had for cheap. Somewhere along there I came across some vegetarian peanut/malt homemade candies that were delicious. They are cut half inch thick, about 3 inches long, all different flavors...sort of like nougat, chewy and flavorful. Coffee flavor was really good, as some others. At another part of the street there were also some very good hard bean curd snacks. Again fresh and wholesome. All the simple things that are good in Taiwan..that after coming back to the US I can only get a shadow of from the stale packaged imitations that's in the chinatown store..sigh.
On an aside, I don't know if you've managed to get a subway map in Taipei, yet. If not, you need to get one, just to see what a Chowhound oriented country it is. It reminds me of the idea we have back here in NYC for the 7 train, where we could easily annoted each station with names of restaurants/stands for food.
My next trip, if I could, I would bring a handful of friends to just go eat our way around Taiwan....
Enjoy while you're there!