it is true, no car rental office at the punta cana international airport.
you can rent a car on several companies like avis and they bring you the car to the airport.
once you arrived right where you leave the baggage claim where all the reps of the touroperators are hanging out a person will hold up a sign with your name or with the name of the company where you rented your car.
they let you sign the papers on the table of the bar over there.
i had several customers renting cars like that, it always worked without problems, but it always takes time, so don't arrive in a hurry.
several times the "full covarage" the clients thought it is included has not been so, they had to pay extra for that right there on the airport.
best way to get a rental car would be to call a company from your hotel, you will find contact numbers near the frontdesks of all hotels, the nice guys of the guest services will give you a hand with that too. the car rental companies deliver your car to the hotel.
have fun
Mike