According to a video produced by the New York Times, the FBI will not process any gun permits under a government shutdown. Some additional research on this issue provided an opposite opinion:
Law enforcement activities are designated as essential services and do not shut down even if the Congress and President fail to agree on funding legislation. The NICS system will continue to operate.
I suppose that we will have to wait to shut down the government in order to see what the government will “allow”. Since my business is a licensed FFL, I will contact the FBI on Tuesday morning to find out what the administration will decide if the shutdown occurs? I wonder is pResident Obama thinks this part of the FBI is an “essential service”.