According to this news article, as of 2005 you could buy a car in California without a driver's license. I'm looking around the California DMV site and I don't see anything different there, although the title application form has a spot to put your license or ID number. The bill of sale form doesn't. You can always call them.
Some states (MN) don't permit this, which is kind of dumb since you may be blind or in a wheelchair yet want to buy a car so your caregiver can drive you around. One of those state ID cards would probably suffice though.