(Many thanks to Linda Julien for providing information obtained from the Reservation office!)
Climbing permits are free, but are not given on Sundays and holidays due to the high volume of visitors in the Chasm. To obtain a permit, arrive at the Chasm area with your climbing gear, and locate a ranger. After checking your gear, the ranger will issue you a permit.
My assumption is that your permit is good for the day, but check with the ranger when obtaining your permit, or call the office and inquire.
Information update, taken from this newspaper article:
"Climbers are required to have a climbing permit signed by a park ranger before attempting to scale the rocks at Purgatory. The permit is free and contains a waiver holding the state harmless for any injuries that might occur while climbing. The permit also requires that climbers "possess adequate skills and the proper equipment to pursue this potentially hazardous sport." Climbers are not allowed to place pitons, bolts or zip lines on the rocks.
Because of the high volume of visitors on Sundays and holidays, climbing permits are not issued on those days."