3 of the 4 go-kart venues Go Karting USA tracks in Minnesota host birthday parties, based on what each venue publishes and what visitors describe. Most of them cluster around Minneapolis.
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.
Stockholm Karting Center, Pro Kart Indoors, ProKART Indoor Racing — all listed below with ratings and locations.
Minneapolis (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 4 operating tracks in Minnesota — see the full Minnesota guide for all of them, not only birthday parties venues.