I am not aware of any PRS-style matches on that side of the state (if that's what you're looking for) but there are various Appleseed matches near there. Up north a couple hours is Camp Perry, then like ithers have said, East is Rayner's (monthly LR matches, monthly pistol, 3-gun matches, and a fall sniper match).
Thunder Valley in Coshocton has matches every now & then and is hosting a PRS match next month.
Southington is up near Cleveland/Akron and has a couple matches coming up too.
Edit to add:
Dayton --> Rayner's 2.5 hrs, $20, steel to 1,000yd
Dayton --> TVP 2.5 hrs, $30, steel to 1-mile
Dayton --> SHC 4 hrs, $20, steel to 700yd
Rayner's is usually open all the time, call.
TVP announces open range days
SHC is probably open for shooting the least as they do training and rock crawler stuff too. Best to call them.