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Did you bother searching online for locations with FedEx, UPS, DHL etc?
On Fedex.com, click Locations on top, enter Los Angeles International Airport to search for, and then click Filter, under Additional Services you can find "Hold package at location for pickup" to find an office just outside the airport open until 9pm on Saturday that offers hold for pickup services. There's reliable public transit there as well: a shuttle to the green metro and then the green line.