There are two - one is on the left side of the resort. Pass the check in desk in the lobby, take the hallway on the left, take the elevator down one floor, and the laundry room is up the hallway a bit at the next "alcove".
The second is out by the pool - leave Mara and walk to the left, follow signs for Simba's cubhouse. This laundry is convenient if you are hanging out by the pool, but is a little more banged up than the other one. One time a hose blew off the wall and water was spraying everywhere (looked like an episode of I Love Lucy, with all of us trying to catch it - none of us thought to simply turn the machine off - too many pina coladas!).
I also bring a roll of quarters (I pick it up at the bank before I leave), and a baggy with drier sheets. I pick up detergent on our way into Disney (town car stop).