While Wyoming seems like anatural destination for bringing Fido along, the national parks are absolutely awful destinations for pets -- they aren't allowed on the trails or many in-park hotels. If your itinerary is focused on national forests and dog-friendly cabin resorts, however, it can be workable to bring a dog on the trip.
Keep in mind that dogs must be leashed at all times on all federal and state lands. However, because many of the public lands in these states have bears and other wildlife, there are sometimes site-specific regulations regarding pets. Aside from regulations, be attentive to your pet's well-being. Just as people need extra water in this dry climate, so do pets. And keep in mind that trails are often rough, and jagged rocks can cut the pads on your dog's feet. Remember, too, that dogs that usually spend most of their time sleeping aren't used to 10-hour hikes up mountainsides.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.