There is a good drop off service in the Plaza just before Plaza Estrella heading towards Bavaro Plaza from Fruisa. It is the plaza where the EPS office and the Drink Bar is. I forget the name. On the second floor on your left going in.
There aren't many laundromats in a country where you can pay someone to wash your clothes for peanuts. Either do that, or there's a decent drop-off laundry service/dry cleaners in Punta Cana Village - Bubbles Cleaners, Tel. 809 959 0357. There are one or two in Ver?n, one near Plaza Genaro and one further up the road (direction of Higuey) called something like El Bosque. You can also rent washing machines by the hour.
It would be a borderline risk. First of all, ask is there a high need, a market for the service ? You would have to have a full time honest employee, if not yourself, to oversee the business. You would be a slave to the business. Self-serve equals you a relying on the client to know how the use and not to abuse the equipment. The public in this country will push the equipment to the limit and then some. You would have to have support to maintain and repair equipment. You will need an continuous water supply and uninterrupted electric. The building would need high security against theft. You would have to market and price the service to reach clients you desire. You need a lot of up front cash to start the business and what is most important, location, location, location.