Hi,
I am told that my great-great-grandfather,
Thornton Provens, carried mail by horseback,
and that it was then called The Pony Express.
The thing is, he did this in Ohio. Is this
possible?

Usually when people talk about the Pony Express, they are referring to the Express established by Russells, Majors and Waddell, which operated from April 1860 to October 1861 between St. Joseph, Missouri to Sacramento, California. There were other horse expresses. I don't know if there was one in Ohio.
Hope this helps