The Midtown Global Market is a cool, unique, interesting place with numerous food stalls offering dishes from practically every continent. It's also a good place to get imported package goods and fresh produce. It speaks to the multicultural spirit of Minneapolis. I went today and the Mexico Tourism Board had musicians playing lovely tunes.
That said, I don't think there's too much for vegans here. The new Salty Tart bakery occasionally has vegan cupcakes and some other vegan baked goods, but you should call ahead to make sure stuff is in stock, as I've been there many times only to be disappointed that they ran out of their only vegan item. Holy Land Deli has some items which look vegan (couscous, veg. curry) but their veggie burger has cheese in the patty. There's a vegetarian dish at the African restaurant and vegetable roti at the Jamaican soul food place.
You can also try a vegetable torta at Manny's Tortas. They build to order, though I found that if you don't speak Spanish, describing what you want could be tricky. Fortunately I do speak Spanish and was able to get a confirmed vegan torta (vegetable sandwich). It is a soft grilled baguette filled with mushrooms, avocado, beans, lettuce and tomato. It sounds like a burrito, but it's much lighter and less dense. But at $6.45 it's awfully expensive considering the small portion size.
Be warned there are at least two large meat counters which is a huge turn-off. At least they're in the corners of the place and not dead-center in the market. On a whole, though, the Midtown Global Market is a hip place to explore different foods, do some grocery shopping or hang out and get a cup of tea/coffee at one of the cafe vendors.