There is Dominican Shuttles which does Ground and Air Transportation with-in the country and a bit outside. http://www.dominicanshuttles.com/
Then there was AeroDomca who used to fly around the country but after a quick search not sure they still exist. Other than that I know of no others.
There are currently NO domestic flights by a regular carrier within the DR. I believe if you wanted to spend the money you could find someone who has a private plane who would take you but the cost probably would be very high. This might be out of Isabel del Torres Aeorpuerta.