Probably most card terminals in shops have a magstripe swipe slot. But unless they get regular American customers the person serving might not remember how to use it, and management might have decided it's not worth the risk accepting such cards.

You won't be able to do anything at an ATM if you don't have a chip on your card.

Cards without chips are much more widely accepted than American Express, though - is that even accepted in many places in the US either these days?