In short, the answer to your question is July 20, 1890. Just keep in mind that the organization of The Way to Rainy Mountain must be understood before the answer can be easily found. The major parts of the book are called "The Setting Out," "The Going On," and "The Closing In." These major parts are also separated into twenty-four sections with each section divided into three separate voices: one having to do with myth, one having to do with history, and one having to do with Momaday's personal story. Your question has to do with the historical voice (always found in the middle of the three).
One particular historical section deals with Aho (Momaday's own grandmother) and her experiences with the Sun Dances in her young life. Aho participated in the last, full Sun Dance in 1887 above the Rainy Mountain Creek along Washita River. It was when Aho was ten that she witnessed the Kiowa people both surrender and disperse during a Sun Dance without being able to complete their important religious rituals. Why was the Sun Dance left incomplete? The soldiers from Fort Sill showed up on July 20, 1890 and stopped all proceedings. This, then, is the date of the Kiowa surrender.