Married Margaret Burger according to the 1912 genealogy. No other information is available for her. On ancestry.com on the Emerson family tree a Margaret Burger is shown as the wife of this Elijah -- that is an error as that Margaret Burger is married to a another (2nd cousin, once removed) Elijah and they have a son Abraham VanHorn Olmstead and move to Steuben Co., NY
Sources
1912 genealogy "Olmsted Family in America"; #4205
Olmste(a)d files of Walt Steesy [Steesy-1]
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DNA Connections
It may be possible to confirm family relationships with Elijah by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: