A few names to google:
Cathy Montesano (her husband has a different last name, but I cant remember it. Something with a T)
Greg Beroza
Lance Karcher/Sarah Darish
Dennis Farrell
Carl Juul Nielson
Bob Perry
Denise Durnin
There are dozens of others too, you just need to do your research to find them.
Mattituck/Laurel Vet (used to be mixed animal, not sure if they still are. John Anderson might be the guy to try)
I think Howard Flynn might be a sheep/goat vet, not sure what else he does.
You can also get on the LIVMA website and go through their directory.
If you have no equine experience at all, your best bet is to start with animal experience. Half the liability is the people who aren't horse-savvy.
Try Pal-o-Mine, Horseability, Long Island Game Farm, SCSPCA/ERC, or rescues like Project Sage. There are wildlife centers in Huntington and Hampton Bays.
You can also try farriers/massage/sports medicine/dentist people to see that aspect of the industry.
In that area, any opportunity isn't going to just fall into your lap, and like someone said, you might be stuck driving a good distance. I never had a problem getting equine hours.