OK, first off, looks like you are stuck with LAX, but there 2 much closer Airports, Long Beach (LGB) and John Wayne/Orange County/Santa Ana (SNA), both just 14 miles away, LGB is my favorite, small and easy access (Just drive down Ball Road, the eastern end of the road ends at the Airport), though limited flights due to its size. John Wayne is a medium size airport but still has one main pickup hub area, like Long Beach.
But if you are stuck with LAX, besides the Disneyland Express Airport Bus, both SuperShuttle and Prime Time Shuttle have special manned pickup points near each terminal (and there are 8 terminals, some do share a stop). Any other Shuttle service has to wait off property until they get a phone call that you have collected your luggage and are ready to pick up. Both Uber and Lyft are allowed to pick up at LAX, though they do add a surcharge, that is passed on by the service to the passenger(s).
For example, a taxi from LGB or SNA is about $50 plus tip, LAX is around $100 plus tip, though traffic waits add to the meter.