If you have the booking number (and of course his name), you may be able to use the “manage my booking” section of the website to see his status. If the flight has departed without him, some airlines will actually show him as “no show” on the relevant flight. Or you may see him having been rebooked to another flight.
If you have his passport number (and full name and birth date, and his consent), you may be able to check his I94 history online, though I’m not quite sure how fast this is updated.
The airport most probably won’t know anything (but you can of course try calling them, you never know...). The ones who would know (but may be hard pressed to give any information for privacy reasons) are CBP (though I’m not even sure it’s easy to reach the local CBP at LAX), the first airline if he was denied entry, and the second airline if he did indeed go through and boarded the flight or was rebooked to a later flight.