You would have to hire a cab. The post office is about a mile, or 30 min walk each way, from the car rental agencies or a nearby hotel, so none of the free shuttles are too useful for getting there. While there are no USPS facilities at LAX, the attendants are various lounges will often take mail for you (AA in T4, DL in T5 and CX in TBIT definitely do this). Before you ask, the reason for the lack of mail boxes and USPS facilities at LAX is simple: bomb threat. So it isn't an oversight, and they won't be restoring service in the next few years.