Renting strollers in the parks can be very expensive. It is $15/day for a single stroller and $31/day for a double. You cannot take these out of the park which can make a long walk back the hotel transportation.
I have also used outside sources for stroller rental. You sign up and pay online, they drop the jogging stroller off at bell services and you have it for the entirety of your trip. I loved this but you do have to drag them onto buses and boats. When you have a full bus at closing, standing with a folded double jogging stroller is a little tiresome (especially for my hubby).
Orlando Stroller Rental & Magic Strollers are 2 of the outside sources I have used or know someone who used them. There are other reputable companies in Orlando, but I don’t have personal experience with them.
If you are debating about as stroller, remember that EPCOT is the distance of a 5k from entrance to World Showcase and back. My 7 year old still needs to be in the stroller for part of the day. I HATE having to carry my 50 pound daughter! Especially in the heat, consider a stroller even if you wouldn’t normally put your child in one to go anywhere else.