Chicago
In the three years I've lived here, I've discovered there's a bunch of great options for people eating both gluten free and vegan. Restaurants like the Chicago Diner and Green Zebra are known not just in Chicago, but throughout the US and offer a great selection of vegetarian and vegan options. Whole Foods even stocks some of the Chicago Diner's vegan cookies and muffins throughout the Midwest. As gluten allergies and wheat intolerances become more known, restaurants are getting a lot better about providing GF options. Chains like Edwardos even offer GF pizza's now, and there are some great restaurants with GF menus or choices like The Sit Down Cafe and Luscious Layers Bakery.
While issues can arise when you need to eat both GF and vegan, there's a lot of places where there isn't a problem. Below is a list of some of my favorite places...
Argo Tea: For a chain, Argo's great (though a bit pricey). They've got lots of options for tea and drinks, a tasty GF vegan chickpea salad, and I've started to notice more GF vegan pastry items too. Argo's also a nice place to study- they'll give you 2 hours of internet with each purchase and there's lots of outlets for computers.
Green Zebra: This is an upscale (mostly) vegetarian restaurant on the Near West Side. It's a fun eating experience with lots of different tastes, small plates, and some good GF options. You often need a reservation, and they sometimes don't seat groups over 4 because of the small size of the restaurant. I went a few months ago and it wasn't as good as I remember it from past visits, but it's still a nice meal out.
Hyde Park Produce: I go to this grocery at least twice a week. Their prices are decent, their produce consistently good, and they've got lots of delicious things like polish pickles, GF mixes, almond milk, and even things from Rose's Wheat-Free Bakery (the seeded loaf, while being very expensive is GF and vegan).
Luscious Layers Bakery: If you ever need professionally made cupcakes or cakes, or virtually anything sweet, check these people out! They're super friendly, deliver for cheap, and have an amazing range of GF vegan dessert options.
The Chicago Diner: Ask for their GF menu! The sweet potato fries are amazing, but be careful if you're really sensitive to gluten, as they don't have a separate fryer for GF foods.
The Sit Down Cafe & Sushi Bar: Ask for their GF menu! If you order sushi, they have tamari (GF soy sauce) which is really nice! They also have a whole bunch of sandwiches available on GF bread and GF dessert options. Last time I checked though, neither the bread nor desserts were entirely vegan. When it comes to sushi, I actually prefer the vegetarian sushi at Shinju Sushi, just down the street from The Sit Down (try the shiitake maki and vegetable maki rolls!). They don't have tamari though.
Whole Foods: Even though my mum calls Whole Foods "Whole Pay-Check", it's still a great place to pick up some hard-to-find items and they have consistently amazing produce. I like Whole foods for their tofu, cheeze, hummos, and alternative-milk selections. They've also got great bulk foods (nutritional yeast, bulk spices, nuts), DIY peanut and almond butter, cereal and snack options, and a fabulous baking section. They've also been stocking more GF vegan options in the bakery section- last time I was there, I spotted some GF vegan desserts from Swirlz Cupcakes.