Found in a Wisconsin biography of Elijah Richmond's grandson, Marvin W. Acker, was a tidbit about Grandpa Elijah. It stated that Elijah Richmond was in the War of 1812 and drove an ammunition wagon (presumably with a New York Regiment).
A guide to researching War of 1812 records at the National Archives can be found here. The NARA guide mentioned series M602 (234 microfilm rolls) which can also be found at the Library of Michigan (Index to Compiled Service Records of Volunteer Soldiers Who Served during the War of 1812). Perhaps I can find evidence of Elijah Richmond's participation in the War of 1812. The NARA guide stated: "The records are arranged by state or federal volunteer unit and thereunder alphabetically by name of soldier."
More work to be done!