What requirements do A&Ps (AMT) have to work on avionics?
An A&amp;P may install avionics per an STC or replace a component, but they may not "repair" avionics in the general sense (solder the circuit board). FAR 65.81 specifically prohibits the mechanic from performing “any repair to, or alteration of, instruments.” It takes a FAA Part 145 certified repair station to work on instruments or avionics.  (Remember that autopilots and other “electronic devices for automatically controlling an aircraft in flight”are instruments by definition.) Appendix A to Part 43 defines the calibration and repair of instruments (including autopilots) as an appliance major repair.

Theoretically an A&amp;P could do repairs if he met all requirements for a part 145 repair station but it is highly unlikely it would ever happen.

The repair station or A&amp;P may re-install the part, fill out the maintenance log, and return the aircraft to service. If the A&amp;P re-installs the unit, the reg's say he must have up to date installation manuals, and it specifically requires that “if special equipment or test apparatus is recommended by the manufacturer involved, he must use that equipment or apparatus or its equivalent acceptable to the administrator” to verify system performance (Section 43.13 (a)). 

Title 47 of the Code of Federal Regulations, Telecommunication Commission, Part 87—Aviation Services, states that a commercial radio operator license is required to repair and maintain all aircraft stations and aeronautical ground stations (including hand-carried portable units) used to communicate with aircraft.

In the United States, neither the FAA or FCC provide a "Avionics Technician" license or equivalent as the military or other parts of the world do. No certification or license is needed to work for a "repair station".

The FAA is always revising rules and regulations such as promoting ADS-B and   no-longer requiring a STC to promote safety.