Is there an official FAA source for time zone of an airport?
Each airports timezone is listed in the Airport/Facility Directory. I think the FAA is starting to call these "Chart Supplements" now though. You can search for specific airports here but it looks like the results are in .pdf format. The value you want is circled in red here:



EDIT: As far as digital sources, explicit time zone information seems missing from the NASR airport (APT) data here. The next best thing is Latitude/Longitude it seems.