Do I need to have a Social Security Number to buy an used car here?

No, just reply to the ad and then deal directly with the seller. The site does not sell cars, it's just a free listing site. Anyone can create a "for sale" listing here, and anyone can reply to those listings. I would assume however, if you buy from a new or used dealership that has a listing here, that you will need official identification to complete sale, including title xfer, registration or temp tags, just as you would at any dealership.

Some dealers will try to force you to give them your SS# but you do not have to.