I've dealt with the NJ DMV on 4 trailers, so I feel your pain. The last time I walked into an NJ DMV office with a Certificate of Origin, the clerk looked at me and said "Just what do you expect me to do with this?" (ugh)
The key as to how easy this will be is going to depend on the weight of your 15SC. NJ will not issue a title for any trailer with a unladen (dry) weight of 2500 lbs or less. If the dry weight is 2501lb or greater, only then is it titled.
If your 15SC has a dry weight of 2500 lbs or less, then this is cake - you want to contact the person you bought it from and have them give you a **notarized** bill of sale with the following information:
Year, make and serial (vin) number of the trailer
Name of buyer and purchase price
Date and sellers signature
If the trailer was previously registered ANYWHERE, the name on the bill of sale must EXACTLY match the name it was registered in. The purchasers name must also exactly match whatever is on their NJ drivers license. The bill of sale must be one page and *that page* must be notarized. Notarized cover sheets are NOT acceptable (been there, done that).
The bill of sale does not need to be fancy. It can be handwritten or typed, doesn't matter. Something like the following is simple and 100% acceptable by NJ:
Bill of Sale
1980 Sunline 15SC Trailer
Seller: John Doe
Buyer: Mike Smith
Purchase Price: $1000.00
Date: June 23, 2010
(signature) (notary stamp & signature)
The above is all you need. Take that into the DMV office and bring your checkbook. You will be paying for the registration fee plus NJ state sales tax on the purchase price. You will not get a title, but it will be legally yours in the eyes of the state of NJ and you will receive a plate and registration card.
If the weight exceeds 2500 lbs and you don't have a title, this is going to become much more difficult. You will still want to obtain the same notarized bill of sale. You may also want to obtain a notarized letter/statement from the seller indicating that they never registered it since they bought it. If this is the case, you may want to sit down and spend 2 hours on hold and talk to somebody in the trenton "special titles" office - they're miserable to get a hold of, but they know how to help.