Hi Margaret! How fun that you and your cousin will be enjoying the magic of Walt Disney World Resort together. I completely understand that you would want to be able to enjoy the theme parks up until the last moment when it is time to go to the airport.
If your luggage isn't too big and is within the maximum allowable bag size of 24" long by 15" wide by 18" high according to the
Walt Disney World Property Rules, you could bring it with you to the park and store it in a locker. Just be sure to check the
locker sizes in advance.
Magic Kingdom Park and
Epcot have jumbo lockers that may be large enough to store your bag. All bags will be inspected prior to entering the park.
You don't mention how you plan to get to the airport, but I am assuming you are planning on using a taxi or rideshare service. Perhaps you could store your luggage where you are staying, and stop by to pick it up on your way to the airport. Since you and your cousin are renting a car together, another idea would be for you to keep the car since you are leaving later.
If none of those options will work, try doing an internet search for luggage storage in Orlando. I was able to turn up the names of a couple of local companies that do luggage storage and transfers.
I hope that helps! Please let us know if you have any additional questions. Enjoy your visit, and time with your cousin!
Anne Marie