FAR 91.3(c) states, "Each pilot in command who deviates from a rule under paragraph (b) of this section shall, upon request of the Administrator, send a written report of that deviation to the Administrator."
In order for this regulation to apply, two circumstances must have occurred during the flight: There was an in-flight emergency, and the pilot in command deviated from a regulation in order to meet that emergency. If these conditions have been met, then FAR 91.3(c) gives the FAA the authority to request that the pilot in command submit a written report of the deviation, and the pilot is required by the regulation to comply with that request. The regulation does not specify who in the FAA may make this request, but we would expect that the request would probably come from air traffic control or the flight standards district office. But, let's be clear here. There is no requirement to submit a report unless the FAA asks for one. If the FAA does not ask that you submit a report, even if there was an in-flight emergency and a regulation was violated as a result of meeting that emergency, there is no regulatory requirement to send a report to the FAA. And, if such a report is requested, there is no time requirement specified within the regulation within which to submit the report to the FAA. We presume that the FAA would interpret the regulation to require that the report be submitted within a reasonable amount of time after the request is made.
FAR 91.123(d) states, "Each pilot in command who (though not deviating from a rule of this subpart) is given priority by ATC in an emergency, shall submit a detailed report of that emergency within 48 hours to the manager of that ATC facility, if requested by ATC."
Before FAR 91.123(d) may obligate a pilot to submit a report to the FAA, three conditions must be met: there was an emergency, the pilot in command was given priority by ATC, and ATC requested that a report be submitted. Note that a deviation from a regulation need not have occurred. If these conditions have been met, then the pilot must submit "a detailed report of that emergency" to the manager of the ATC facility that made the request for the report. Under this regulation, the FAA office that may make the request is identified in the regulation, i.e., air traffic control, and the regulation specifies the time requirement within which this report must be filed, i.e., 48 hours. Note, as with 91.3(c), there is no requirement to send any written report to the FAA unless ATC asks for one.
Fortunately, the chances that you will experience an in-flight emergency are slim. However, if you are involved in an emergency that requires you to deviate from a regulation or receive ATC priority, you may expect that you will be requested to submit a report of that emergency to the FAA. If so, you should obtain competent advice in preparing the report, and at the very least, submit "just the facts, Ma'am, just the facts."