I understand that Royal Caribbean's Vision of the Seas will not be sailing out of Santo Domingo after early April this year. So you probably need to concentrate on other cruises on the links mentioned in an earlier reply.
I know nothing about necessary visas for Dominicans, but there were lots of Dominicans on the 7-day cruise I took in late February. They didn't seem to have any problem about going ashore.
There are 2 different cruises out of Santo Domingo at the moment. One apparently requires a visa for Dominicans - stops in Antigua, St. Kitts, Barbados & St. Lucia. The other one apparently does not require visas - stops in St. Maarten, Dominica, Margarita Island & Aruba.
We just returned from the last mentioned cruise. Lots of Dominicans going ashore, and it didn't seem to be any different than anyone else leaving the ship (then again - I didn't take particular note either)
The last round-trip cruise out of Santo Domingo is April 12-19. Starting April 19, the ship will be doing a transatlantic cruise leaving Santo Domingo going to Harwich, England. This is a one-way cruise, as they are re-positioning the ship.
We talked to a cruise consultant on our cruise and asked him if there would be cruises out of Santo Domingo next year - he said that as of right now, the schedule has not been finalized, but they should be publishing the 2010 cruise dates by about April 1, 2009