Both the Best Western Disney Springs across the street and the B Resort and Spa have guest laundry facilities. I'm not sure the location in the Best Western, but the B Resort and Spa website says:
"For your convenience, we have a 24-hour laundry facility on the property. Laundry facilities are located behind the hot tub area and are $4.00 per washer or dryer load."
https://www.bhotelsandresorts.com/b-resort-and-spa/amenities
I'm not sure if a key card is necessary to access the area. It's only a 3 or 4 minute walk over to the B Resort (it's on the same side of the street as the DoubleTree Suites). If it's open access, I wouldn't hesitate to use the facility.
If you can't access that facility, ask the DoubleTree Front Desk to call down to the Hilton Lake Buena Vista to arrange for you to use their laundry. The DoubleTree and Hilton operate under the same hotel group, so it would be natural for those two properties to help each other out.
Or, just ask the DoubleTree Front Desk to call the B Resort and Spa or the Best Western. I worked for many years in hotels. In one hotel in which I worked, our biggest competitor was directly across the street. Despite being competitors, it wasn't unusual for one hotel to help out the other one.