This is a gold medal (actually bronze) awarded to an exhibitor at the 1904 Louisiana Purchase Exhibition, also known as the St. Louis World's Fair. We are currently working under the assumption that this came from my grandfather's things and not my grandmother's, but we could be wrong. In either case, it wouldn't have originated with either of them; my grandfather turned six in February of 1904 and my grandmother was born that March.
I have found, on the Internet Archive, a copy of the list of exhibitors at the fair. It is a very long list, the book is over 1300 pages long, it will take me some time to get through.
Since my mother doesn't remember ever hearing any family stories about the exhibition, I don't know that we'll ever solve this mystery with any degree of certainty, but I hope to at least make an educated guess.