There are many restaurants in New York that accept credit cards. The ones that do will usually have a sign outside that lists the cards that they accept. If they don't accept credit cards, there will usually be a sign outside that says "Cash Only".
Most payment processors will accept credit cards. All online ones like PayPal, Google Checkout or Authorize will accept credit cards. For offline ones you have to look at their terms of business to see if they accept creditcards.