Posting this information for any who might need it as I spent quite a bit of time looking for a place to do laundry in Positano.
There is a coin operated laundromat (in Italian it is a "Lavanderaia") right past Hotel Villa Franca and right across the street from the entrance to Hotel Il Gabbiano. The street is Vialle Pasitea...I don't recall the # but Hotel Il Gabbiano is at #310, so it will be close to that. It is called L'Arcobaleno Lavanderia.
It is clean and they are friendly, but fair warning it is not cheap. It was 6 EURO per load to wash or dry, so several loads will easily run you up to $30. We ended up washing only, then hang drying our clothes to save time and money.