There is no "permit" -- for a Regional road racing license, you have to go through school to earn the initial license, then complete a couple of actual races.
SCCA - Sports Car Club of America has the initial info...and note that going through school requires you bring an appropriately-prepared car with you. Your best bet there is to rent one that already has passed initial inpections and has a good logbook.
For Solo or PDX, there's nothing required beyond the driver's license any of us monkeys can obtain.