I don't know about LAX...
We have used these for our travels in and out of SNA.....
Avis Rental Car (2004)
Budget Rental Car (2008)
Super Shuttle and they were fine, no problems (2010)
24-7 Ride (2012) They were fine, no problems. Their price was best when I checked around. We took 24-7 Ride from SNA to the GC.
Yellow Cab Co has a FLAT RATE for Disney. Call the hotel, bell service or guest service can tell you all the info about it. I think their rate from LAX to Disney is $85. Disney to SNA is $35. Nick P was our driver. He was a fine driver, again no problems. I took Nick's business card and when we're in town, needing a ride, I will call him to be our driver.
If you like to schedule everything, then I'd go with 24-7 ride or super Shuttle.
I prefer not to schedule. I like the instant availability of Yellow Cab. Their flat rate plan isn't much different than the shuttle services either.
As far as CAR SEATS.... When I checked back in 2010, I never found a transportation service that provided them.
They said they would not be able to provide transportation if the kids were not in approved car seats. We had to pack our own along for that trip. We just used a bungie cord to hook them to our carry on bags.
Good luck getting all the plans set for your Disney trip