No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Where you hoping that getting that loan increases your score prior to car purchase?
If that's the case, pay the loan below 9% and you'll get all your points back.
Once you have car loan, self lender is redundant and waste of money
I think the OP is suffering from the "no open installment loans" hit.