At Santo Domingo airport,
Car Rentals from Budget has a 6-7 passenger mini-van on their website at SDQ airport for $515 for a week Jan 10-17
Avis Rent A Car, Inc. has a 9 seat full size Hyundai van for 9 people for $516 Jan 10-17
You can check other car rental sites.
Are you looking for a van AND driver?
Problem with just about any rental agency van that you'll drive is that if you fill it up with passengers there won't be any room left for luggage.
I know that companies exist that will rent larger shuttle type buses with drivers as our company rented them a few times in Santiago when we had large groups of visitors. Just wouldn't know where to suggest you look in the capital.
If you speak some Spanish, the yellow pages may help.
PaginasAmarillas.com >>> PUBLICAR
ps: I drove several years in DR and do now whenever I visit. However, I personally wouldn't attempt driving a van loaded with 9-12 passengers. Distractions from passengers, poor visibility, and stopping distance with that weight isn't something I'd want to deal with in DR traffic - be it the city, highways, or mountains.
If I had that many people, I'd go with a van AND driver, or split the people up amongst two mini-vans.