Why isn't "pay with card" accepted? Can't find any reason why that should be wrong?
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.
We presently require "with a card".
Perhaps it is possible to elide the undefinite article, but could you show us some evidence for this possibility? How did you find out?
As the ngrams shows, it is certainly not used too frequently.
Well, the ngram shows that it would make most sense to put there "Can I pay by card", obviously.