You can do either. They can charge it to your room if you'd like or you can pay cash. Just tell the person at the valet stand which you'd like when you go to pick-up your car.
Make sure to get a receipt. You can then use valet at any of the other Disney resorts without having to pay (the daily fee) a second time. I don't know if the valet at the Swan and Dolphin are linked to the other resorts (I'd be surprised if they were), but you can certainly "hop" between the Disney resorts.