There will be 6 of us (4 adults and two 1 year olds) and the cheapest I've found for a rental van is $500 and unfortunately that's just not in our budget. Between the airport shuttle and occational taxis if need be it will still stay well below the cost of a rental van, rental car seats (I can't imagine trying to lug ours for gate check along with all our luggage and stroller) and what parking at the parks would be for the week. So though I would love to have a car...it just doesn't seem to be in the cards for us.
Also try when renting a car to put in 1 extra hour. Pick up at 10:00 am and drop off at 11:00 am. I find that the price actually is cheaper in most cases. What about renting two cars instead of a minivan? We rented a car for $135 for the week. It would be worth it if you had two drivers and really don't plan on using the car that often. You may want to bring your car seats from home. I know it can be a pain, but the rental companies charge a lot for car seats. Your children are little and you'd rather be safe than sorry.