Hi there C! Welcome to the Disney Parks Moms Panel.
It sounds like you are one very busy momma! I have three children that are 5 and under so I get how
Planning for Preschoolers and Little Ones may be a bit overwhelming. While I do not think that you are the only one that may need a quad stroller, I think that it is not something that can be rented. I was looking for one stroller to hold all three of my little ones for our last trip but I came up short when I looked at all the Stroller Rental Companies in Orlando.
If you have two adults in your party I would highly recommend renting two double strollers. My favorite place to rent from is Orlando Stroller Rental. You can rent from them for the length of your stay. They will drop off the strollers to your Resort hotel before your arrival and then at the end of your stay you will take them to Bell Services and the company will come and pick them up. I have found that the strollers are clean, very easy to maneuver and fold, and the company has always been a breeze to work with. The best part is they are extremely affordable in my opinion. On our last trip we rented two City Mini strollers for our 6 night stay and they cost $70 a stroller for the length of stay. Please remember that this price is subject to change without notice.
Since your kids are still so young I want to encourage you to take advantage of the Baby Care Centers located in each of the Theme Parks. These centers are a great place to change diapers, feed the kids, and just cool off! Lastly make sure to schedule in down time and naps for both the kids and the adults! Traveling to the Walt Disney World Resort with little kids in tow can be a bit tiring (I know from experience) but when you make sure to schedule in this down time I have found my vacations go much smoother!
I hope that you have a magical first visit!
~Heather~