There is a Walmart a few miles down the road from me... I don’t have to drive down there and confirm that they do not sell them. They don’t. I guarantee that if someone ordered one like that, it would be canceled, due to shipping restrictions.
Need to remember that 49 states (probably less, but emphasizing NJ is an anomaly with it) don’t consider BB guns firearms (similarly, black powder guns). A department store like Walmart isn’t going to be my go-to legal advice on BB guns. They probably don’t get the inquiries through the website enough for them to write it in that BB guns are a no go for NJ stores. Years back, Walmarts and K-Marts did sell BB guns, as well as shotguns and .22s... but most let their FFLs lapse. Sports Authority had the same situation, and there was one in Eatontown that kept theirs when all others stopped selling guns (until the chain closed completely).
Go to a Dick’s in NJ, their BB guns are behind the counter (airsoft is not, being they are considered more along the lines of paintball guns). You go in, they do a Certificate of Eligibility and a NICS (judging you have a FID), you walk out with your BB gun. Buy any ammo (exception being shotgun ammo), they log your FID, put ammo in a locked plastic container, which is opened by the register as you check out.
Listen, I’m very glad people do want to help the OP, but BB guns are considered firearms under NJ law. That means they are subject to NJ 2C:39 statutes, which can be easily be assessed. I’m giving him what the law is within NJ, without having to read pages of statutes, which are complicated for a specific reason. NJ’s constitution does not give citizens any rights regarding firearms, so 2C:39 is what you have to look at for legality from the NY border, and between the Delaware River, Hudson River, and Atlantic Ocean. There are a handful of AG decisions to consider (like the odd one saying Shockwave/TAC-14 firearms are legal, and not requiring a pistol permit for), but nothing changes the fact that it is illegal for a non-resident to stay in NJ with a BB gun, without abiding by the exemptions. As described by the OP, it is illegal unless he owns the property he is staying at. Whether he is going to get caught is another argument, but in regards to NJ law, it would be breaking the law.
The OP is an adult, and can decide whatever he wants in his travels. If he was worried enough to post a thread on this, I’d just suggest to take the Turnpike (I95) to the Delaware Memorial Bridge, and within a few exits, you are at a Cabela’s where you can buy a BB gun, and not have to worry about NJ transportation laws. Plus, DE has no sales tax.