We ended up using Lyft on our recent trip instead of taxis. Lyft had a promotion so what we did was use my husband's app on our way to the airport at home and he got $10 off that ride. We then used a referral from him to get $10 off our trip to Anaheim from LAX by creating an account for me. He then got a promotion for $5 off his next 4 rides so we did a two small rides in Anaheim, one to LAX, and one from the home airport using Lyft Line to save more money. There may be terms that limit you from using them at the same time but its worth looking into what they can offer you as you would save on multiple cars vs a van. I also believe others have mentioned if you use Uber X you can request a specific car. Someone here also posted a link right before I left for a car service with reasonable rates, that would ensure you have the car or van you need which may be a better option as you will not want significant delays with small children. We discovered that due to conflicts with taxi car drivers Lyft and Uber cannot pickup or drop off in certain areas, so we still had a wait but it was better than the taxi lines at LAX.
On that note our driver from LAX to our hotel had a nice conversation with us and he himself was a father with two small children. One of the drivers we had in Anaheim was nice and the other was older and very quiet. Since you have a wide range of people working for these car services you never know, but compared to NYC taxi drivers they were very charming and kind