Amy, the lawyer that moderates here may take awhile to give you the definitive answer.
There is likely nothing that addresses this, nothing that prohibits it. The government has rules that private schools must follow, though. The reality, as amp says, is that the system is not effective in ensuring that Dominican students are in school. So a foreigner that's "homeschooling" would fall off the radar. You wouldn't be required to register etc. I'd venture to say if you tried, you'd be met with one blank stare after another.
Most posters will likely confirm this.
The real question, although not a legal one, will be your support group. Your best bet is to do what you are doing now--reaching out online. You will likely find some missionary families that are out there, on their own.
If you are enrolled in an online program, I'd suggest you continue with that. You'll need to be prepared to be on your own curriculum wise. Public libraries aren't common, but if you have flexibility in where you can live, an area like Las Terrenas that has an international expat community might be right for you. The older your children are, the more you'll be dependent on the internet.
My disclaimer: I'm not a lawyer, I didn't homeschool, and I don't live in the DR!
BUT because this is the
legal forum, I quickly skimmed the Dominican law that relates to education. The only two things that I saw that might relate are that parents have the right to decide for themselves the type of education for their child (I think it refers to which school) and a paragraph that says distance learning is acceptable (but that is together with paragraphs that refer to adults): LEY 66-97 Ley General de Educaci?n
http://www.oas.org/juridico/spanish/mesicic2_repdom_sc_anexo_7_sp.pdf
There are a few posters who have more experience with the regular school system, and hopefully they will chime in with their thoughts. You may want to repost the question about the logistics of homeschooling (as opposed to the legal) in another forum for more answers to that part of your question.
Good luck to you.