WINDOW ROCK, Ariz.—Carmelita Lowe stared down at a script inside a makeshift theater on the Navajo reservation and did her best to sound like a cartoon fish speaking in an ancient tribal language.

“Hi, I’m Dory,” Ms. Lowe said softly in Navajo, hoping to fill the role of the chatty regal blue tang in a new Navajo-dubbed version of the animated movie “Finding Nemo.”

The...