"....I have a copy of his death certificate, but where can I gain any other documents connected to this young man...."
Is your interest confined to his military record?
Find My Past has details of his death and gives his birthplace as "Kingswood, Bristol". I think I have found him on the 1911 Census but Rootschat rules preclude posting details of the 1911 Census..
This leads to the 1901 Census (RG13 2350 8 5)
Joseph Gay, Head, Marr, 39, Shoemaker
Esther Gay, Wife, Marr, Shoe Maker
Children - George 16 (Tram Conductor), William 12,
Earnest 8, Ethel 3, Elizabeth 21, Frederick 18 (Tram Conductor),
Ann Jeffries, Aunt, Single, Cotton Weaver
All born Kingswood, address Hanham Road, Kingswood
Was Walter christened Walter E(a)rnest? I can find him as Walter in 1911 but there is no Walter in the 1901 family - only an Earnest with the same year of birth as the 1911 Walter
FreeBMD has a birth record for a Walter Ernest Gay, Keynsham, Mar 1893, 5c 588 (Kingswood comes under Keynsham Registration District)