Obviously "card" means "credit card", thus "I can pay by credit card" would be accepted as well. Reported on Nov. 22. 2017.
신용카드 would be the direct translation of credit card.
Card/카드 would be more generic for debit and credit cards combined.
But I agree your suggestion could be accepted as well.
Also if you look at the situation of buying 'you' is more likely to be used.