You said "private car". How are you defining that? Private, as in not for hire (like visiting Aunt Betsy who lives in Lakeland), or private, as in not sharing with anyone, like a limo or car service?
Rental cars companies rent seats to go in the cars they rent out. They do not rent them to people who are not also renting a vehicle. Nursery equipment rental companies, like A Baby's Best Friend, or Baby's Away, will rent to anyone (including Aunt Betsy), and will deliver the seat to your hotel/home. If you are taking a car service or limo from the airport to your resort, they *should* be able to provide a carseat for your use whilst in the car, as long as you make it a point to reserve ahead, and be specific about what size you need.
If you need to rent a seat, consider costs carefully; the rental often ends up costing as much as the price of a lightweight travel seat (such as a Touriva, which will run about $45 at Wal-Mart.)
