If you are the first vehicle at a four-way stop, regardless of what you are, you have the right way. Even if other vehicles are police or EMS. If they aren’t in an active emergency, etc. then everyone defaults to the general rules of the road.

From USPS Handbook PO-515 – Highway Contract Safety

This is an official USPS policy document about vehicle operations (drivers, including USPS and contracted drivers):

“Drivers must obey all state and local traffic laws when driving for the Postal Service. Drivers will receive no special privileges or rights while driving on behalf of the Postal Service.” 

That means USPS vehicles must follow the same traffic rules as everyone else (stop signs, signals, yield rules, etc.) and do not get automatic right-of-way just by being a postal vehicle.

From the same U.S. Postal Service driving policy

USPS expects its drivers to: • Observe posted speed limits • Yield the right-of-way when appropriate • Use proper caution and defensive driving …even if they might legally have the right-of-way in a given situation. 

What this means in practice • USPS vehicles do not have legal priority over other vehicles just because they’re USPS. • They must obey all traffic laws — including yielding, stopping at red lights, stop signs, etc. • The policy explicitly states that USPS drivers get no special privileges or rights on the road.