I think I found the Chapman family in 1900 living in Windom, Lewis Co., WA.
William Chapman, head household, b. MS, Oct 1859, widower, farmer.
Nancy A., mother, b. MS, May 1830, widow.
James A., son, b. TX, Jan 1880, single, farm laborer.
William T., son, b. TX May 1881, single, farm laborer.
In 1910 William Sr. and family was living in Cowlitz, Lewis Co., WA.
William Chapman, b. MS, 54 yrs old, 2nd marr, farmer.
Elizabeth Chapman, wife, b. IN, 45, 2nd marr, had given birth to four children who were alive in 1910.
William T. Chapman, son, b. MS, 29, single, farm hand.
Ray F. Taylor, stepson, b. OR, 20, single, farm laborer.
Mormon Taylor, stepson, b. WA, 16.
In 1920 Ray and William T were living in Skookumchuck, Thurston Co., WA.
Ray Taylor, b. WA, 30, single, partner, timber faller log camp.
William T. Chapman, b. TX, 38, partner, timber faller.
A William Chapman died 5 March 1936, 60 yrs old, in Yelm, Thurston Co., WA. Death Note=Found 9 March 1936. Source: Washington State Death Index, image #1198, document #=74., www.digitalarchives.wa.gov/.
I don't know if this is your William Chapman or not.
You can request a copy of his obituary from a volunteer at Random Acts Genealogical Kindness. Go to www.raogk.org, read instructions, choose Washington, choose Thurston Co.
Washington State Library, www.secstate.wa.gov/library/ask.aspx . They will search through the newspaper microfilm for 60 minutes at no charge & email to you what they do or don't find. Takes about two weeks. Successful Searching!