Jane, How wonderful of you to take time to visit Grandpa! I hope he's feeling better soon.
With all of your supplies and 2 little ones, I would certainly consider keeping the car for the week. This way, you can secure the car seats one time, and you won't be moving them in and out several times. I also liked having a car when my kids were that age for going to the store if I needed to go for diapers, formula, or whatever surprises pop-up!
When renting a car, check
www.mousesavers.com for rental car discount codes. Some of these codes can save you a lot of money. I like to rent from a car company that has their cars right at Orlando airport so I don't have to take a shuttle to pick up my vehicle. I find Dollar usually has competitive rates.
Since you are staying on property at Saratoga Springs, parking at the 4 theme parks will be free. At check-in, you will get a parking permit which allows you to park at your resort, and you will get you admitted to the parking areas for free at the parks. You can still use the bus/transportation system around Disney if you like.
Have a nice vacation with your family!