Can I get official recordings of ATC audio?
In the specific situation of a pilot who is the subject of an FAA investigation, the Pilot's Bill of Rights requires that air traffic data (which includes audio recordings) be provided to the pilot. They would be expected to make the request directly to the FAA. There is a procedure for this:

To ensure that relevant air traffic data you are seeking from a government contractor are obtained, you should provide the following information:
a. Your name and your airman certificate number;
b. other relevant information (aircraft heading, altitude, call sign, transponder code, etc.) that will help the contractor locate relevant data; 
c. the name/location of the contractor facility that you believe has air traffic data;
d. the date and local time of day when such data relevant to the your flight operation were generated;
e. the name of the FAA inspector, investigator or other FAA official (e.g., an air traffic controller who advises you about a possible pilot deviation) who notified you that you are under investigation;
f. the airman's phone number or email
The individual should provide that information to the FAA by email to the following address: AirmenDataRequest@faa.gov.

The instructions include a warning about timeliness:

Depending on the type of data, they will normally be stored from 5 to 45 days and then they are destroyed or discarded in the normal course of business

I don't have any information about the general case, though I imagine it depends on the facility and their workload.