Terminal 1 (All Airlines but Southwest)
360° Gourmet Burrito
Vegan options: You can get a salad or burrito with smoked tofu. It appears from their website that all of the sauces are vegan.
Vegetarian options: Any of the above with cheese and sour cream. Quesadilla options. Breakfast burritos with egg and cheese.
Website (For main menu - selections within the Oakland Airport may be more limited.)
There are also your run-of-the-mill snack bars with pre-made sandwiches and salads, which typically offer few, if any vegan choices. Breakfast is especially limiting with little more than fruit and bagels with jelly.
Terminal 2 (Southwest Airlines)
Vegan options: Mesquite Grilled Veggie Burrito
Vegetarian options: Above plus Huevos ranchers (Breakfast), Vegetarian Burrito (can be vegan without cheese), Quesadilla options
Website Note: I have contacted Andale to verify ingredients (namely that no chicken stock or lard are used in the cooking process) but I have not heard back.
California Pizza Kitchen
Vegan options: Tomato Basil Spaghettini, Thai Linguini, Grilled Vegetable Salad, Field Greens Salad. Some dishes may be vegan by requesting no cheese.
Vegetarian options: A large variety of meatless pizzas and pastas.