Monument Valley can be found on the border of Arizona and Utah. The park is owned and run by the Navajo Nation.
Flying can be difficult with a dog and often not permissible if they aren’t a service animal. But if you do go that route the closest airport to Monument Valley is Flagstaff, Arizona and it’s 176 miles away. Not exactly close. Other major airports are much further and include Phoenix (320 mi), Albuquerque (324 mi), Salt Lake City (380 mi) and Las Vegas (400 mi).
We are from Phoenix, Arizona so we just drove from Phoenix to Monument Valley. It’s about a 5 hour drive.