I showed them my NJ firearms ID — which apparently does not fulfill their requirement.
As I wrote in another reply on this thread, I’ve purchased ammo at Gander Mountain in Middletown before, but have never had to show any kind of permit/ID to buy that ammo.
I've purchased at the same store and been asked for my NJ FPID. I've also had to show my FPID at the Walmart just down the street from there.