Downtown Phoenix is pretty small, but, walking in the heat can be brutal in July. There are several recent threads on this topic. Also, keep in mind the light rail is nearby. One of the contributors, here, writes a nice food blog with pictures of what's available along the rail line.
Within walking distance of the Wyndham, some ethnic and/or vegetarian options are as follows. Keep in mind that walking distance is a relative term. A breezy half mile stroll in January can be an endurance test in July.
Thai Elephant -- ethnicity is obvious; most dishes can be made with tofu instead of meat; just a block from the Wyndham
The Breadfruit -- healthful Jamaican with some vegetarian options; nearly half a mile from the Wyndham
Sens -- Asian tapas with some incredible tofu dishes. Right around the corner from the Breadfruit
PastaBar -- handcrafted pastas, most of them meatless. Next to Sens.
Moira -- sushi with some meatless options. A few blocks east of Sens/Moira
Verde -- Mexican. Most dishes are meaty, but there the tacos al fresco are a nice summery meatless entree. Near Sens and PastaBar
For the cluster of restaurants at the north end of Downtown, you may wish to walk the half mile, but do with some water and, ideally, a hat. Walking in 110-degree weather takes some preparation. Alternatively, take a pedicab if you can find one in the heat.
For additional ethnic and vegetarian options, you'll find even more choices if you take the train east to Tempe. Some of the stations with the highest concentrations of that type of restaurants are Mill Ave / 3rd St., Veterans Way/College, University/Rural, and Apache/Dorsey. You can view restaurants by station or type of food on the site that John kindly mentioned. If you want to take the train all the way to its eastern terminus in Mesa, you can also visit Mekong Plaza, an Asian-oriented shopping center that is home to Vietnamese, Thai, Filipino, and Taiwanese restaurants.
Phoenix AZ, Phoenix, AZ
You could take the light rail or a cab to Mrs. White's Golden Rule Cafe. Dynamite soul food, and they advertise that the veggies are cooked without meat (though you might want to confirm this). Its $13 and change for a meal, and I can pretty much guarantee you will be full.
Mrs White's Golden Rule Cafe
808 E Jefferson St, Phoenix, AZ 85034