Can the terms of a car purchase be changed after the fact?

A contract may not be unilaterally changed--changed by one party without the consent, or agreement, of the other--after it has been agreed to by both parties. The first key issue here, however, is whether the contract itself stated you were approved for credit--in which case they'd have to give the financing--or whether the contract did not in fact provide that--in which case they do not.

Assuming the contract did not obligate the dealership to provide credit, then the next key issue is whether they only reason you signed the contract was that it was represented (promised or stated) to you that you would get credit. If that was the case, then there are several grounds on which you should be able to get out of the contract (get you payment back): one would be mutual mistake, whereby you and the dealer each thought you'd be approved for credit and only entered into the deal on that basis, but in good faith; another would be fraud and/or uniltareral mistake coupled with fraud, if the dealership either had no idea whether you'd be eligible for credit, or even thought you would not be, but nonetheless claimed you get financing with no basis for the claim, just to get you to sign. So if the provision of financing was something promised to you and was a reason why you entered into the deal, if they can't come through financing, you should be able to get out of the contract--though bear in mind that if the dealership refuses to let you out and tries to enforce the agreement against you (such as by keeping your payment), you will have to sue to recover your money.