Yes, you can fly a drone in Punta Cana. You do not need a permit or a license if it weighs less than 2 kilos. Flying drones is regulated by the Dominican Civil Aviation Institute (Instituto Dominicano de Aviación Civil or IDAC).
Follow some recommendations before bringing your drone to the country. It is recommended (but not obligatory) to request a permit from IDAC (http://www.idac.gob.do) before bringing your drone to the Dominican Republic – it will facilitate your entering the Dominican Republic and going through customs clearance.
If your drone weight is above 4.4 pounds, you will need a permit issued by IDAC. Start applying for a permit at least one month before your trip.