Since you mention you're waiting for a silencer, I assume we're talking about a Form 4? If it's an intrastate move like you mentioned, then it's no big deal at all, you can still pick the silencer up from your dealer when the tax stamp comes in. The ATF almost always mails the tax stamp to the dealer, but we have seen a few instances where the tax stamp got mailed our customer erroneously, so it's a good idea to make sure your mail is forwarded to your new address.
Now, if you had moved interstate then things would be a lot more complicated. If you no longer lived in the state where your dealer is located, then you wouldn't be able to pick up the silencer once the tax stamp comes in. NFA items are "other firearms" on the 4473, and -- just like handguns -- the dealer can only transfer those to in-state residents. The moment you move out of a state, you are no longer a resident of that state for the purpose of transferring firearms under federal law (unless you still maintained a second home in the original state and were actually living there during the time the firearm was transfered to you, but most people don't have multiple homes in different states so I won't go into detail on that one).
In the case of an interstate move, the original Form 4 would be cancelled, the dealer would submit a Form 3 (dealer-to-dealer transfer), once the Form 3 clears the dealer would ship the silencer to a dealer in the customer's new state, then a brand-new Form 4 would be submitted . At some point there would be a $200 refund issued for the original cancelled Form 4, but I hear that takes a while.
Kind of a pain, but that's the only option. We had that happen to a customer recently who had an unexpected move across the county during a wait for a tax stamp.
