I regularly send firearms that are C and R eligible on my Class 3 license through the USPS as priority mail. The gun must be C and R eligible.
It is not legal to mail a handgun through the UPPS using your C&R. Long arms may be mailed by anybody as long as they are in compliance with GCA 1968 and whatever other laws are applicable. Per USPS (bolding is mine):
"11.1.3 Authorized Persons
Subject to 11.1.4,
handguns may be mailed by a licensed manufacturer of firearms, a licensed dealer of firearms, or an authorized agent of the federal government or the government of a state, territory, or district, only when addressed to a person in one of the following categories for use in the person's official duties:
a. Officers of the Army, Coast Guard, Air Force, Navy, Marine Corps, or Organized Reserve Corps.
b. Officers of the National Guard or militia of a state, territory, or district.
c. Officers of the United States or of a state, territory, or district, whose official duty is to serve warrants of arrest or commitment.
d. USPS employees authorized by the Chief Postal Inspector.
e. Officers and employees of enforcement agencies of the United States.
f. Watchmen engaged in guarding the property of the United States, a state, territory, or district.
g. Purchasing agent or other designated member of agencies employing officers and employees included in 11.1.3c. through 11.1.3e. "
http://pe.usps.com/text/dmm300/601.htm#wp1099709
My C&R license specifically states: THIS LICENSE DOES AUTHORIZE THE HOLDER TO ENGAGE IN THE BUSINESS OF DEALING IN
ANY FIREARMS.