Short answer - Individuals or Trustees cannot, Dealers can.
The reason you can ship a riffle or shotgun by the United States Postal Service but not a Short-Barreled Rifle or Short-Barreled Shotgun is because of how Pistols are defined by the USPS.
The USPS defines a SBR or SBS as a Pistol and not a Riffle and as such does not permit them to be shipped by individuals or Trustees of Gun Trusts. See below
Pistols, revolvers, and other firearms capable of being concealed on the person (for example, short-barreled shotguns and short-barreled rifles) are defined as handguns. The following definitions apply:
- Pistol or Revolver. A pistol or revolver is a handgun designed to be fired by the use of a single hand.
- Short-Barreled Rifle. A rifle having one or more barrels less than 16 inches long is defined as a short-barreled rifle. This includes any weapon made from a rifle (by alteration or modification) resulting in an overall length of less than 26 inches.
- Short-Barreled Shotgun. A shotgun having one or more barrels less than 18 inches long is defined as a short-barreled shotgun. This includes any weapon made from a shotgun (by alteration or modification) resulting in an overall length of less than 26 inches.
For more details on this see the Mailability page of the USPS website.