The Postal Inspection Service (external crime) and Office of Inspector General (internal crime) are the two law enforcement agencies currently attached to the Postal Service. They are, of course, armed and have investigative and arrest authority.
In years past some regular postal employees were armed in an official capacity while performing certain duties. Mail clerks on the old Railway Post Office and Highway Post Office systems, which were discontinued in the mid-70's, often carried holstered snub nosed .38 special revolvers while sorting mail on the moving trains and buses. Many local post offices were issued revolvers (the ones I saw were .45 ACP) which were kept in the safe and were primarily used for escorting trucks carrying high value Registered Mail articles to the area processing center. This practice was also discontinued in the mid-70's.
Of course, things were different in "the old days". Unofficially, many rural mail carriers had guns in their vehicles. Many of these old retired carriers like to tell stories about favorite hunting spots on their routes. There's even a story about a post office housed in a little country store in the 70's with an FFL...
Everything changed in 1986 with the shooting at the Edmond, OK post office. That was the first of several post office shootings over the next few years and spawned the term "going postal". Since then, possession of a firearm on postal property or on duty by a postal employee is cause for immediate dismissal.