I played Palmetto on a couple of occasions over the last few years during Masters week. The process was, and may still be, to simply call the club and reserve a time. It think, as I remember, they fax a form which you sign with a credit card guarantee.
Since the leaders at the tournament get off after lunch, it is a great way to kill the morning before going to Augusta.
The golf course is very old school and terrific---MacKenzie from around 1900.