Hi, Larry! My son is 7 and is a tough little guy, but he is more than happy if I offer him a stroller when we are at Disney! It is a LOT of walking, so if they can rest their legs for a few minutes it really helps.
You can go without it, and if your princess starts to get a bit weary, show her the rental strollers and ask her if she would like to use one. I am sure she will let you know her feelings about it!
Strollers can be rented by the day, so perhaps when in a smaller park like Magic Kingdom you can go without one, then when you go to Epcot where there is a lot more walking and less shade, rent one there. Click
HERE for stroller rental information.
See how it goes as it may be different each day. Good luck!