Unless something has changed, you are out of luck--even the post office drop boxes have been sealed up. You will need to take a cab to the nearest drop box off site and it is a pain in the as* to do so. I tried to have one of the nice lounge people take my letters/overnight packet and they said they are no longer allowed to do so becuase of liability concerns. I ended up pleading with a travlers aid person in the united terminal and they took care of it for me only becuase they took pity on me.

I can only hope that the above situation has changed since last year but until proven otherwise, do not count on mailing anything out of the LAX airport.