I found a John W. Vermillion, b. Jan 1857, in the 1900 census (ED 156, Chickasaw Nation, Indian Territory). John was in dwelling 119, and my widowed g-grandmother Cordelia A. Vermillion Lee (soon-to-be Landers) was a moderate distance away in dwelling 178.
John's wife was Susan, b. Jan 1857, but they had only been married 6 years, so she must not have been his first wife, and could not be the mother of his children. Children in the household were Walter, b. Jan 1878, Ira, b. Sep 1881, John, b. Aug 1883, and Reava(?) Vermillion (daughter) b. May 1885, and Eunice Criner, b. Dec 1893, listed as a boarder. Does this sound like your gg-grandfather?
As others have mentioned, I think your John W. was the son of John H. Vermillion and Mary Smith. John and Mary were also the parents of my g-grandmother Cordelia.
I hope this is helpful to you. I'd like to exchange info with you, because I'm pretty sure we're cousins.