They do have it at a few resorts in DR, the last one we stayed at, Barcelo Bavaro Palace in Punta Cana, had all you can eat lobster at the seafood restaurant, where you order it from the menu, and they bring you 3 halfs at a time.
Occasionally, when Lobster is abundant and cheap, some of the upper class AI will have a one day sea food event, which may include lobster tail halves.
Actually, let me correct myself, if I remember right, the last time I witnessed such a happening, it was at the rater moderately priced VIVA resort on Coson Beach (Las Terrenas).
The sad part on these events however is the behavior of so many of the all-you-can-gobble-up crowd.
Even when more often than not, these lobster tails are of questionable quality and mostly overcooked so bad they taste like styrofoam.
I dont think you are going to find lobster in any AI in puerto plata, but you can find all you can eat lobsters in severals rsorts in PC.
the best resort for lobsters is Majestic colonial and Elegance. all you can eat lobsters at seafood restaurants everyday.
Barcelo bavaro beach used to have all day long lobsters, but it has changed.
Ocean Blue on friday night at the buffet.
Bahia Principe, Palladium, Melia Caribe, mostly in gourmet restaurants.
Forget it in Riu. their food quality is going down in last years. spent few days in Riu Palace punta cana (which is the top of riu resorts), and had just the lobster flavor in a lobster bisquet.
you will be lucky if you get lobster in any resort in the puerto plata area.