When we were a family of 5 we used a town car service. We were staying for 5 days and didn't have tons of luggage, but we did have a stroller besides the suitcases. I don't remember a shortage of space, but make sure that the town car service knows how many people are in your party so the town car trunk doesn't have a lot of things in it before you add your luggage (I flew with my son in February and the town car trunk had a lot of things in it--a booster seat, which we then used, a full sized carseat and other assorted things. We didn't have many things so it didn't matter to us, but I remember thinking that it could be an issue for a family. Or you can just pay the small extra charge for a van.
T&B