Some UCs have key lockers. EWR and IAH are among them. I don't have a comprehensive list, but am unfortunately pretty sure that LAX does not. At least, I've never seen any, and with the current construction and craziness at the LAX lounges, I'd find it to be highly unlikely.

Unfortunately, to my knowledge, there is no on-site luggage storage of any variety at LAX.

If you're leaving the airport, depending on where you're going, your most likely chance of success is to find a large chain hotel nearby your destination, tell the Bell Desk that you checked out that morning already, and now want to store a bag for a few hours. Most of the time it'll work.

Good luck!