There is no legal requirement that a borrower has an SSN or ITIN. If you can convince someone to lend you money, your lack of an SSN or ITIN should not be a legal barrier. However, depending on how much money you are borrowing, you likely will find few if any who would lend to a person without a verifiable credit history.
Moreover, you likely will have to produce a driver's license and proof of insurance before the car dealer will let you drive off. Without an SSN, you likely cannot get a driver's license.
You will need to have a driver's license and an insurance card to leave the car lot. In theory, it is possible to get a loan without a SSN, however, the major loan companies will not give you one without a SSN.