Hello Stacey! Ah, the days of having a "wild toddler" are very fresh on my mind and it has been quite awhile. Naps were a "must do" for us and, yes, many of our visits did not make much time for quick runs back to the resort. So, let us start with some ideas that worked for me.
First, if you are bringing your own stroller and know that your child has been able to take a very comfortable nap, I would definitely bring it along. Toss in a very light blanket to drape over the top since the sun will be shining and it will be hot. I always brought along a very small light pillow just to add to the comfort.
Now I, personally, have always chosen to rent a stroller from the parks. I always rented a double stroller so that I could make a padded bed and my child had more room to stretch out on. The strollers are very easy to maneuver and provided a very smooth and comfortable ride for my child. I would bring along two extra small pillows to place in the "dip" from the seat to the leg room. Yes, I know that three small pillows (two for the "dip" and one for the head) does sound like a lot to bring along, but the children were very comfortable for the ENTIRE day. Some naps went as long as 2 hours. YEA!!! Again, bring a light blanket to keep the sun out.
I would recommend a powered fan to attach to either stroller. The water fan bottles are also a big plus, but if you want to save on money, purchase these outside the park.
I hope this helps as you begin planning for your big day. Have a magical day!