Welcome to the Dis! You are correct that there is no "free" transportation from the airport to your resort. You can choose from a number of different options to get your family to the resort and around WDW.
You can rent a minivan from one of the rental agencies and others here can really help out with more info on costs. I have no idea how much a minivan rental goes for since we have never chosen that option.
You can hire an agency to take you round-trip to your resort and back to the airport -for your group you would probably use a van instead of a towncar. There are lots of different companies you can choose from including
FL tours, Yourride, Tiffany Town Car, etc... The car (van) services charge a flat-rate per round trip transfer. I think a round-trip van tranfer for most services would be about $80-$100 depending on the company. Some offer luxury vans, others are more like larger vans with rows of seats (like a church van or something).
Mears shuttle service is another option that some people here choose and they provide transfers via a shuttle bus (sometimes a large van, other times a full-size bus). You pay them X amount of dollars per person for the transfer. They do also have towncars to provide transfers but again, I do not have personal experience with those either-sorry.
I have used Mears for airport transfers and we had the unfortunate experience of having our shuttle involved in a minor accident after dropping guests off at a hotel (this was in Miami for a cruise transfer). They made us wait for more than 2 hours before they would let us off the bus to get into the hotel and arrange other transportation to the port. We almost missed our departure and others did miss it and had to be flown to the next port. Mears is not high on my personal list because they were not equipped to handle this type of situation and because I think there are better, more reasonable options. I have used Tiffany Town Car and would probably choose yourride or fl tours for our next trip because of the decrease in service I experienced from Tifffany.
Whatever you decide, just research your options and make the best decision for your family. Good Luck!
