From BATF website
(B11) May a person who is relocating out-of-state move firearms with other household goods?

Yes. A person who lawfully possesses a firearm may transport or ship the firearm interstate when changing his or her state of residence. Certain NFA firearms must have prior approval from the Bureau of ATF, NFA Branch, Washington, DC 20226, before they may be moved interstate. The person must notify the mover that firearms are being transported. He or she should also check state and local laws where relocating to ensure that movement of firearms into the new state does not violate any state law or local ordinance. [18 U. S. C. 922( a)( 4), 27 CFR 178.28 and 178.31]
link
But that is pertaining to shipping via a mover, not sure if that means you can mail it or not.