I'm sure there are laundromats, but I don't know where they are.
I don't use the washers/dryers at Disney (or anywhere else), but I did go to the laundromat a lot when I was younger and they seem to be pretty standard.
Cheapo washers do not get clothes as clean as the ones you might buy yourself. You can wash something 3 times in them and it just won't get that stain out...then you put it in a good washer and Voila! Stain is gone. But they do a decent enough job for typical washing.
If you sweat a lot in the clothes, I'd wash them twice and dry them 2 or 3 times on a lower setting. The "hot" setting on commercial dryers can scald your clothes...you get these yellow/brown burn stains on them that cannot be removed.
So, the laundry facilities in the hotels will surely be as good as any laundromat, but the laundromat (with all its washers and dryers) will probably be faster.
You can send the clothes out to be professionally laundered through the hotel or a laundromat, but they may also shrink the clothes or damage them (buttons missing, bra hooks bent, color changes, etc.)
Good luck! (bump)