Definitely make sure you know the exact model year and VIN of your trailer before you try to apply for a title. Unlike cars/trucks, the VINs on these things can be anywhere and most police/DMV employees wouldn't know where to find them.
Like Gene said, a bill of sale is a good thing to have from them. Plus, if you have any problems, you may actually need the previous owner to come down to the DMV or police department with you in order to clarify everything for them.
If you need any help finding the VIN/model year, feel free to let me know.
If you still have issues getting a replacement title, you may have to go to the NY DMV with the previous owners and basically have them apply for a lost title but instead of issuing them a new one, they issue it to you instead.
2007 T-286SR Cherry/Granola, #6236, original owner, current mileage: 9467.8 (as of 5/26/19)
1997 T-2653 Blue Denim, #5471
1979 12 1/2' MC, Beige & Avocado, #4639
Past Sunlines: '97 T-2653 #5089, '94 T-2251, '86 T-1550, '94 T-2363, '98 T-270SR