Does in-flight Wi-Fi require regulatory approval?
The FAA touches upon it in this letter


  Current FAA regulations require an aircraft operator to determine that
  radio frequency interference from PEDs is not a flight safety risk
  before the operator authorizes them for use during certain phases of
  flight. Even PEDs that do not intentionally transmit signals can emit
  unintentional radio energy. This energy may affect aircraft safety
  because the signals can occur at the same frequencies used by the
  plane’s highly sensitive communications, navigation, flight control
  and electronic equipment. An airline must show it can prevent
  potential interference that could pose a safety hazard. The PED ARC
  report helps the FAA to guide airlines through determining that they
  can safely allow widespread use of PEDs.


The operator is responsible for determining that the unit does not create any interference they link off to this document which is pretty verbose and outlines a lot of the info you may be looking for on this front.  

This document elaborates a bit more on the regulations 


  Aircraft with an FAA-approved system— such as an Onboard Mobile
  Telecommunications System (OMTS), Wireless Fidelity (WiFi), airborne
  access systems (AASs), or Network Control Units (NCUs)—are considered
  PED-tolerant for PEDs used with the installed system. If an aircraft
  model has demonstrated tolerance for both transmitting and
  non-transmitting PEDs, the operator may allow PED use during all
  phases of flight on this aircraft model.


Aircraft need to be shown to be PED tolerant for Wifi units. You can find an AC covering whats involved in that here.