Purchase of a car is a business transaction inititated by the consumer.
Permissible purpose to pull the consumer's credit report arises if the business transaction gives rise to an associated "legitimate business purpose" for the dealer to review your credit standing as part of the transaction.
If you are paying by way of cash or certified funds that involve no seeking of credit from them or no reasonable chance of the funds not clearing, there is no legitimate purpose for them to review your credit as part of the transaction. Thus, they could make no request for your credit report, be it hard or soft.
However, if you wish to review credit they might offer, they then would have a legitimate business purpose. The prospective creditor would also have a legitimate purpose for reveiw of your consumer initiated request for credit.
The FTC has also issued a staff opinion letter that payment by personal check would also constitute a legitimate business purpose, as review of your credit history might reflect on the risk of the funds clearing.
It thus depends upon the details of your business transaction with them.
