I'd been contacted off-line about this before it was posted here.
The medals are definitely not from the IC4A. The most common use of "O" in 19th century medals was for the word "order," as in various cities' AOH Games, the Games of the Ancient Order of Hibernians.
But a check of various meets in the NY metropolitan area of the 1880s and '90s showed no such meets that matched the initials on the medals.
I later found out the medals were from a collection in Northern California, which leads me to believe they are from an intercollegiate association in Oregon.
Someone might need access to newspapers from Portlant to get a better fix on this, however.