There are quite a few pet friendly motels in Williams and a kennel. The Kennel is run by the Grand Canyon Train so I don't know if you have to have a reservation on the train or not. Here is the website.
Here is a website with pet friendly hotels in Williams
Most people disagree with me and I understand that and respect that. . Most people advise not to take dogs. We have been there several times without a dog and it has been easier, but it can be done. The biggest hindrance is that you cannot get on the shuttle to any of the stops on the west rim out to Hermits Rest. We have a small dog that we have taken to the canyon often. You can still enjoy the canyon with your dog if you are considerate. We were there less than two weeks ago and Visited the Village are first - where the hotels and restaurants are. Obviously dogs are not allowed in hotels and shops. Hubby stayed outside while our friends and I went into gift shops.
You can buy sandwiches at the marketplace or at Bright Angel Fountain located outside the gift shop and restaurant. They sell drinks and sandwiches that you can eat anywhere outside.
We left the Village area and then drove to all of the stops on the east rim - Rt 64E. They are all beautiful and you can take you little dog with you. The last stop is Desert WatchTower then you can continue on to Cameron and head back to Flagstaf on 89A