When it came time to pay for my current car I gave them my AmEx. (I had previously checked to make sure they'd authorize the charge). They balked. I pointed out, nicely, that their merchant agreement requires them to take it. They offered to give me LoJack, already ordered, free if I paid cash. Since the $695 price was worth more than the miles to me but their own cost for it was less than their credit card fee would be, it was a win-win for both of us. I wrote them a check.
So, even if they won't take (or won't want to take) a credit card, it can pay to suggest it.