According to the LAX website, postal services are not available in any of the terminals at LAX. It has been a while since I have seen a post office in any airport terminal in the U.S., which I assume owes to a combination of the declining use of postal mail in general and the heightened sensitivity to possible terror attacks through packaged bombs or anthrax.
You might be able to leave postcards or envelopes with the staff at an airport lounge, although the willingness of staff to take mail seems to be variable. I was not able to find anything authoritative as to purchasing US postage at the airport. If there is a news kiosk or bookstore in your terminal, you can ask.
In any case, the nearest U.S. Post Office is a distance away at 9029 Airport Boulevard, across the street from the Hertz on one side and Avis from another. The Avis shuttle would be slightly more convenient, for what it's worth, but it would still be very time-consuming to do on a layover.
For shipping packages, there are also appears to be a FedEx location in the Los Angeles Airport Marriott (5855 W Century Blvd), and another in the Westin Los Angeles Airport Hotel (11221 S Hindry Ave), both with limited hours. The nearest UPS STore is in a strip mall at 8939 S Sepulveda Blvd. A taxi will be most convenient for these locations.