If they aren't requiring it, why are you asking more questions? Submit your FPID and place your order.
If that's what they require then just give THAT to them without stirring up a hornets nest that doesn't need to be stirred up.
Doing a quick internet search I was only able to find the below. Since this is, or may be (not sure if it passed) , a NJ State law I don't see how they could require out of State ammo sellers to submit that information. If the online sellers you use don't ask about it then I wouldn't worry about it.
Handgun Ammo Regulation
This requires ammunition dealers to keep a record of who buys ammo and then provide that record to the state police.
Congratulations, New Jersey. Black market types all over the state are looking forward to this new and exciting business opportunity as people go across state lines to buy up ammo, drive back, then sell the ammo to those who don’t want to show up on those records.
I called TargetSportsUSA the other day to ask. The person assured me that they will ship to NJ with valid FPID, but I'm not sure if that person was fully updated on this specifically. I sent an email to get something in writing but have not heard back.