8 of the 13 go-kart venues Go Karting USA tracks in Ohio host birthday parties, based on what each venue publishes and what visitors describe. They're spread across 2 metro areas, so most of the state is within a reasonable drive of one.
Party bookings work differently from walk-in racing: expect a reserved room or area, a set number of races per guest, and a deposit. Weekend afternoon slots go first, so start calling two to three weeks out — and confirm the height rule for the youngest guests before invitations go out, since it varies venue to venue.
Lorain Ohio Kartplex, K1 Speed - Indoor Go Karts - Canton, OH, Full Throttle Adrenaline Park, Cincinnati: High Speed Go Karting, Axe Throwing, Paintball, VR, Rage Room & Group Events and 5 more — all listed below with ratings and locations.
Columbus (2), Cincinnati (1).
Two to three weeks for a weekend slot, more around school holidays. Party rooms and group race slots are reservation-based at essentially every venue, and afternoon weekend times fill first.
Go Karting USA lists 13 operating tracks in Ohio — see the full Ohio guide for all of them, not only birthday parties venues.