What are the ages of your kids? If they are young enough to be in car seats in the car then they should be in car seats on the plane. While it can be a challenge getting the seats through the airport, your flight will be much more enjoyable with the kids buckled in their seats instead of climbing on your lap, or worse, up and down the aisle. While most airlines allow children under two years of age to travel without a ticket, this means that they don't have a designated seat to themselves and are referred to as a "lap child" since that's where they'll be sitting, unless you are lucky enough to be on a flight that isn't full (going to Orlando, I wouldn't bet on that) Turbulance can be extremely dangerous and even deadly to small children sitting on the laps of other passengers. Those harnesses that you see in some catalogs that are "designed for air travel" generally aren't allowed during landings and takeoff.
Many airlines offer discounts for children under two, and my experience has been about 50% off the adult fare.
If you were to rely on the rental car company to have a van waiting for you with integrated seats, what would you do if they didn't? My experience has been that often what you reserve is not what you ended up driving.
I don't recommend checking car seats on with plane along with your luggage as they can be lost or suffer mistreatment at the hands of the baggage handlers.
My DD has been traveling since she was 6 months old and I have had her car seat everytime (she is 5 1/2 now). I strap her car seat to my carry on suitcase that has wheels and transport it through the airport that way. I also recommend that you preboard, to allow you enough time to get the seats secured on the plane and then get the kids in the seats. I found most flight attendents quite helpful with this when I've traveled alone with my daughter.
Check your child's car seat, there should be a sticker on it with writing in RED that the seat is approved for airline use. The same height and weight restrictions that apply for auto use apply for airline use.
HTH