Quick answer
Мо́жно ка́ртой?
mózhno kártoy — two words that work in every shop
The options
Why ка́ртой
The means of payment takes the bare instrumental: ка́ртой, нали́чными — with no preposition at all, exactly like е́хать авто́бусом.
Cashless Russia
Contactless payment is standard almost everywhere, including markets and small kiosks. Foreign cards, however, have not worked in Russia since 2022 — plan for cash if you travel there.
What you hear back
Typical mistakes
Мо́жно с ка́ртой?
Мо́жно ка́ртой?
No с — the bare instrumental carries “by”.
Мо́жно плати́ть ка́рту?
Мо́жно плати́ть ка́ртой?
Instrumental, not accusative.
Check yourself
Which is correct?