What is the procedure for getting permission to operate an aircraft within a pop-up VIP TFR?
Usually when the President visits LA or San Francisco there are one or more inner rings and at least one outer ring. The inner rings are usually 10 nm in diameter and you can’t enter them under any conditions. The outer rings, on the other hand, are able to be flown into if you are on a VFR or IFR flight plan and in continuous contact with ATC. Here’s a portion of the TFR at Martha’s Vinyard.


  FOR OPERATIONS WITHIN THE AIRSPACE BETWEEN THE 10 NMR AND 30 NMR
  AREA(S) LISTED ABOVE, KNOWN AS THE OUTER RING(S): ALL AIRCRAFT
  OPERATING WITHIN THE OUTER RING(S) LISTED ABOVE ARE LIMITED TO
  AIRCRAFT ARRIVING OR DEPARTING LOCAL AIRFIELDS, AND WORKLOAD
  PERMITTING, ATC MAY AUTHORIZE TRANSIT OPERATIONS. AIRCRAFT MAY NOT
  LOITER. ALL AIRCRAFT MUST BE ON AN ACTIVE IFR OR VFR FLIGHT PLAN WITH
  A DISCRETE CODE ASSIGNED BY AN AIR TRAFFIC CONTROL (ATC) FACILITY.
  AIRCRAFT MUST BE SQUAWKING THE DISCRETE CODE PRIOR TO DEPARTURE AND AT
  ALL TIMES WHILE IN THE TFR AND MUST REMAIN IN TWO-WAY RADIO
  COMMUNICATIONS WITH ATC.


If you are departing or landing at an non-towered field, you need two radios (or an Garmin SL30 radio) to monitor ATC and communicate on the CTAF. 

I have done it lots of times VFR into LA and it really isn’t much different than getting flight following.

I don’t have any experience with departing from a non-towered airport into the TFR, since the last time I entered the TFR the approach end of the runway at Catalina Island was in the 30 mile ring, but the departure end was not. So I don’t know for sure but from talking to the people at the FBO and reading the TFR, I think you need to open you flight plan either by phone or on the radio (Hawthorne radio would be who you call and the frequency depends on where you are in the LA basin), contact ATC (in this case LA Center, again frequency varies) on the ground before you take off, then make your normal calls on the CTAF. Then contact ATC again when you are airborne.