Answer
A foreigner can buy a car before obtaining permanent residency, but a passport alone may not be enough for the entire process. Identification of the parties, completion of the transaction with an escribano, registration, an address for documents, insurance and a legal payment method will be required.
The right to own a car does not mean you have the right to drive it. Separately check how long you may use a foreign driver's licence, the exchange requirements and the required licence category. The insurance company also sets its own list of documents for the driver and owner.
Before transferring the money, choose an independent escribano and send them your passport and the seller's and car's details. They will check the registry, debts, restrictions and whether the car can be registered under your status. Do not buy a car on the basis of a receipt alone without completing the official transfer of ownership.