am i able to pay with a credit card (at this store)?
may i pay with a credit card (at this store)?
I agree with Kaskade's comment. Ability vs Permission. In the case of the credit card example, you are able to use 払うことができますか because the store may not be physically able to process your credit card (lack of a credit card ...
They are not equivalent. The difference between the two is that of ability vs. permission. In English, both of these concepts can be expressed using "can", which is the reason why Google Translate offers the same result here. So these two sentences actually mean:
(1) Am I able to pay by card?
(2) Am I allowed to pay by card? / May I pay by card?