It does not seem to be used at all https://books.google.com/ngrams/graph?content=pay+by+card%2Cpay+with+card%2Cpay+with+a+card&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cpay%20by%20card%3B%2Cc0%3B.t1%3B%2Cpay%20with%20a%20card%3B%2Cc0 Are you sure it is supposed to be correct?
I would have expected the right answer to be "to pay by card" but "by" wasn't displayed at all, so I chose "with". Found out, that it is possible to use it, though not very common. But my answer "May I pay with card" wasn't accepted, I was told there needed to be "May I pay with a card". 100% sure.