Every DVC resort has laundry rooms, and yes, they are free. At dedicated DVC resorts (like OKW and SSR), all the laundry machines are free. At resorts that have a DVC component and a hotel component (like the Contemorary, Wilderness Lodge, Beach Club, Boardwalk, Animal Kingdom Lodge), you need to make sure you're in the correct laundry room. The laundry rooms intended for hotel guest do require quarters.
At one time, some of the machines in the DVC laundry rooms had coin slots on them. You could push them in without putting in any quarters, though. I'm trying to remember if I've seen any like that recently.
The laundry room is on the 4th floor of the villa building right next to the main elevator. Last time I was in there, it had two sets of nice high efficiency machines.
There's a vending machine to purchase supplies if you need them, or you can purchase it at the Marketplace.