Will you have a microwave? If so, you can buy a rotisserie chicken at Safeway (check the label to see if it's still gluten-free). Grab a bag of self steaming micro veggies, salad, and potatoes. I did this in Williams, but at the time we were in an RV. There are a few restaurants in Williams, but not sure how many are gluten-free. You can always order soft tacos (corn tortillas) and ask about the meat and possible gluten. Don't eat any fried tacos or chips because of cc issues (usually)
If you want to eat out go to the El Tovar Hotel in the park for plenty of gluten-free options. You must walk around this historic hotel when you're on the south rim. It was built by the Fred Harvey Company (remember, Judy Garland in the movie "The Harvey Girls"?) Folks used to disembark on the old Santa Fe train for food and accommodations. The hotel is still run by the Fred Harvey Company. Besides the great dining room, there is a café which also has gluten-free options.
Have fun! Bring water and DON'T HIKE DOWN TO THE CANYON DURING THE DAY IN SEPTEMBER! If you must go to the bottom, take a night hike and bring plenty of water, and wear headlamps and bring flashlights. Sturdy boots, pants are a must since rattlers are out, along with scorpions and tarantulas. It's an experience you'll never forget!
If you're going through Kingman when it's time to eat check out In-n-Out. Dedicated fryer for fries. They'll wrap your burger in lettuce (protein style) and tell them you have an allergy. It's written on the ticket and they'll grill it on a separate grille and change their gloves. Stop by the visitor's center there as it's all about Route 66. It's well worth it and you can even bring your dog inside. There's an ghost town off US 93 (coming from Hoover Dam to Kingman) call Chloride. It's a fun stop and we at lunch there and didn't get glutened but it's not as good as eating at In-n-Out considering the options. If you have time, visit Oatman off old Rt 66 (follow the signs) another Ghost Town famous for their wild donkeys/mules. Plus, you'll pass Cool Springs which inspired Radiator Springs in the movie "Cars".
Have lots of fun!