As weird as it may sound, Bartlett Dining Commons is one of the better places in Hyde Park to get a vegan meal, even if you're not affiliated with the University of Chicago. Food is served a la carte at different "stations" and many items are made to order (so they're easy to adjust, according to your dietary restrictions). The Dining Commons itself does not feel like a dining hall (probably because it used to be a gymnasium), so eating there won't feel incredibly awkward if you're an outsider. That said, I recommend that you check your table before sitting down to make sure that it doesn't have a name plaque (unless you want to enjoy your meal with a group of students who live in the corresponding dorm).
The Harvest Station is completely vegetarian and always has a number of vegan options. Sometimes, the Harvest station even has vegan cookies, brownies and/or cake slices.
The other station that is definitely worth visiting is the Kosher Deli. The people working at the Kosher deli can make things like vegan grilled cheese sandwiches (using Tofutti deli slices) and bagels with Tofutti cream cheese.
There is also a fridge with sushi (including inari and vegetable sushi - both vegan), and a few other stations have dishes that can be made vegan (ex. the East/West stir-fry station). The ice cream freezer often has sorbet and there are always "juice" boxes of chocolate and vanilla soy milk in one of the beverage coolers.