Yes, you can compost grapes.
Grapes rot down incredibly quickly in a compost heap (or in a fruit bowl if you’ve forgotten about them). Throw them on your heap, stalks and all.
(It’s away from the subject of composting really but if you’ve got a load of grapes set to go off and don’t want to sling them, freeze them instead. They’re like little mini-sorbets when frozen and can be eaten as snacks without defrosting.)