If you’re feeling adventurous, then maybe public transportation? The Metro 460 line starts at
Disneyland and connects to the Metro C light rail line to the Aviation/LAX station at the airport. Then it’s their airport shuttle transportation to all terminals.
Cost varies depending on whether it’s a day pass or just cash fare. But it will require their “TAP card” which is an interagency fare card. There are no discounted fares for children unless they’re free (under 5?) except for children attending local schools. The card is a $2 one time fee. The fare should be $2.50 for the 460 bus and then a free transfer (within 2 hours) to the C line. Might even be $1.75 by June, when Metro is totally revamping their fare system.
https://www.google.com/maps/dir/Disneyland+Resort,+Disneyland+Drive,+Anaheim,+CA/Aviation+/+LAX,+Los+Angeles,+CA+90045/@33.8818112,-118.1457641,11z/data=!4m9!4m8!1m2!1m1!1s0x80dcd7da84d77583:0x7efd02b2f50ccd51!1m2!1m1!1s0x80c2b6b8e1edfad1:0x71d96422aabc0d0b!3e3!5i1
Not sure where to get a fare card though that might be doable. There are some places near LAX that sell them at machines, and supposedly they can be purchased on board most 460 buses. I tired but was told by a driver that he was out of them but he let us on board with the promise that we would get them in downtown LA along with a day pass used that day.
For for a family of 4 it would be less than $20 and public transportation doesn’t require child seats.