Because The Way to Rainy Mountain is a history of the Kiowa tribe of Native Americans, it doesn't really lend itself to a list of characters the way that a work of fiction would. The reason is that Momaday's The Way to Rainy Mountain is not a work of fiction. The organization of the book can help you to understand this, too. The book has three major parts: "The Setting Out," "The Going On," and "The Closing In." These parts are about the creation, the heyday, and the basic end of the Kiowa tribe. The book is further divided into twenty-four other sections. Each of these sections has three parts: one having to do with myth, one having to do with history, and one having to do with personal experience. The "characters" in the mythological parts vary greatly from the "characters" in Momaday's personal experience.
In regards to characters, however, I would say the most important character is Aho, Momaday's own grandmother. She experiences many important events of the Kiowa tribe, most specifically the Sun Dance. Another important character is the strange creature called Tai-me, which is half mammal and half bird. This creature speaks to the Kiowa and becomes a focus of worship during the Sun Dance. There are many other characters (especially in the mythological part) that have minor roles, such as "the Mole," who provides medicine, and "the Crow," who gives warnings to the Kiowa. Other animals are "characters" that are important to the Kiowa's migration and life, such as dogs and horses and buffalo. Even the soldiers at Fort Sill become important characters who end up dispersing the tribe during the last Sun Dance.