Can you ask for napkins in a coffee shop?
Yes, you certainly can. That's the right word to use.
Napkins, also sometimes known as serviettes, are used for wiping your hands, mouth etc at mealtimes. Washable napkins are made of cloth, such as linen; disposable napkins are made of paper.
The difference between a napkin and a tissue? While paper napkins are a food-related product, paper tissues are a personal/health/hygiene product. You use tissues to wipe tears from your eyes, wipe your nose if it runs and to catch sneezes. Tissues are generally softer than napkins.
I hope that helps.