Please note it would be illegal for a foreigner to work for a a business without having a work visa or residency in the Dominican Republic. In your case, a work visa would seem to be the best option.
Your question is far too general. In what career are you looking for an internship?
When I was young, an internship was basically a volunteer position where you received knowledge and training. In return you gave up your time to require this knowledge. It was a win/win situation for everyone involved.
I find great humor in your inquiry to get a paid internship on a Caribbean Island. We had several words (mostly derogatory and insulting) for this type of request years ago.
I would share them with you but I have been disciplined in the past for insulting people with limited intelligence.
The best I can do is wish you good luck while I go get a cocktail and share this with my wife.
Real estate. I think you can work an unpaid internship DR (you're right pay may be complicated but under the table who knows). The issue with a work visa is all about money.