As a last resort you can enter on a 60 day Visa Exempt stamp if you do not have ahistory of staying in Thailand. When your 90 day visa comes through you will need to leave the country (day trip to cross the border) and then re-enter on your 90 day visa.
The only document that you need to buy sa car is Thai Baht. To register it in your name you will need a Cetificate of Residence from Immigration (or a yellow housebook if you have it and your passport. The seller (if Thai needs the vehicel registration document and his Thai ID. If the seller is not a Thai person, he will need the vehicle registration document, his passport and, I think, a Certificate of Residence.
Fact. I have seen on this site, after the change to 60 days, where many people were pulled aside and questioned on only their second Visa Exempt entry and some were denied entry.
Immigration consider that the Visa Exempt entry is for tourists.If you have already had 90 days in Thailand under the Visa Exempt/extention then Immigration may well consider that you are no longer a tourist and are attempting to stay long term in Thailand and so deny you entry on your second try for a Visa Exempt entry. Immigration has, since the Visa Exempt moved from 30 days to 60 days, questioned/denyed entry to many (but not all) who have tryed for a secomd Visa Exempt entry. Therefore second or third Visa Exempt entrys are frought with risk.