You want
Express by ExecuCar by SuperShuttle. It's a private (non-shared) van service for up to five (5) passengers.
LAX to Disneyland is currently quoting at $72 for 1-5 pax before gratuity or any promo code you might be able to find. The rate is the same for five passengers as for one passenger.
Clean vans, courteous drivers, efficient/timely service.
They also give UA miles.