Any PADI dive shop should have a "Discover Diving" program that is geared exactly for that. You'll likely have to do a quick pool lesson and fill out a medical questionnaire and then you're taken out in a group of 4 with an instructor. I believe the dives you do do count towards the number of open water dives you have to do in order to get full certification.
Another option is you can probably take a crash course here at home if there was one available in the next 14 days. They are usually around 3 days I think if you do a weekend course. Then you could do the open water portion when you're there with a PADI shop. Might be a bit rushed but thought I'd throw that out there.
The "Discover Dive" programs though are usually pretty good for getting a "taste" of diving.
Found this page while googling "PADI Punta Cana":
http://www.godominicanrepublic.com/en/ThingsToDo/SportsAndRecreation/DivingAndSnorkeling/PADICertifiedDiveOperators.htm
Some of the websites listed didn't work for me but this one did:
http://www.pelicanosport.com/english...ba_diving.html
Anyhow, have a good trip.
