I love dim sum and wish there were a veg-only dim sum place in Chicago. (There is a great one in New York's Chinatown.)
That said, it isn't as hard as you'd think to have a great vegetarian meal at a dim sum place that serves meat too. As much as I like Phoenix (the fresh silken tofu is divine-smooth, lightly sweet and so fresh), I'd recommend going to a place like Shui Wah, where you can order off a ticket, rather than with carts. Since there is less veg food on the total menu, you can wait a while for a veg cart to go by. The tickets give you a bit more control. And Shui Wah has a great vegetable rice noodle roll. If you think the carts are essential, they'll work to, just plan on a longer meal and lots of communication about the dishes.
When I go, it's nearly always with meat eaters, and we'll get a variety. Usually we'll order one full sized vegetable dish, like snow pea shoots, off the regular menu and then a variety of dim sum. There's lots to choose from: fresh tofu, rice crepe rolls, dumplings, sesame paste or lotus buns. Yum. Keep in mind that a lot of the cakes and dumplings have shrimp or pork thrown in. If I were you, I wouldn't be shy about getting some meat dishes for yourselves. There will be plenty of food and if you're getting a variety of vegetable dishes, no one will feel left out of the experience.
Have fun!