You may be surprised to know that, not only can it be done, but there are now an abundance of resources to help you: blogs, cookbooks, a website, even a forum. Let's take a look, keeping in mind this list is not exhaustive:
The Gluten-Free Vegan - A wide variety of recipes, including many common comfort foods (mac & cheese, shepherd's pie, muffins and cookies) and a selection of Mediterranean dishes (falafel, tabbouleh, dolmathes). Recipes come in a printable format.
Gluten Free, Soy Free, Vegan - Earthy foods and flavors from a Peace Corps volunteer living in Senegal. Mixed in among standards like pancakes and smoothies, find recipes for pumpkin lasagna, squash tagine, lentil dhaal, or dosas.
Manifest Vegan - An all-purpose gluten-free vegan cooking site. Find recipes to satisfy foodies as well as novice cooks. Also has tags for those who are soy-free, nut-free, corn-free, or refined sugar-free.
See also: The Welcoming Kitchen, Going Against the Grain, Veganza, and Fork & Beans
http://xgfx.org/ - Includes resources, recipes, product reviews, tutorials, tips for baking and for dining out.
http://www.glutenfreeveganlove.com/ - recipes, tips, and lifestyle advice.
Great Gluten Free Vegan Eats, by Allyson Kramer
The Gluten-Free Vegan and Gluten-Free Vegan Comfort Food, by Susan O'brien
The Welcoming Kitchen, by Kim Lutz
Gluten-Free & Vegan Holidays, Flying Apron's Gluten-Free & Vegan Baking Book and Gluten-Free & Vegan Bread by Jennifer Katzinger
http://www.vegiac.com/ - While the activity in the forum ebbs and flows (it seems to be in an ebb stage at the time of this writing), the wealth of information in the previous thread makes it worth creating an account. Find recipes, resources, and advice from others with first-hand experience.
While being both gluten-free and vegan definitely presents some challenges, with the amount of information and support available, it's not only possible, it's now simpler than ever.