You need the stroller, and I wouldn't leave it in the room the first day and see how it goes either. DD (now 7) would not set foot in a stroller after the age of 12 months when she took her first steps. She walked at the zoo, the mall, and anywhere else that required quite a bit of walking and we never took the stroller. She is a high energy kid. WDW was an entirely different situation. She was 2 on her first WDW trip. After a few minutes she realized how huge the place was and happily jumped in the stroller and let us wheel her around. WDW is the ONLY place she has ever used a stroller from the age of 12 months and she used it there up through her 5 year old trip. Plus, if you are trying to get somewhere quickly- like an ADR, show, etc., it is a lot easier to go fast with a kid in a stroller instead of a 2 year old slowly toddling along on those little legs or trying to carry a kid. Also, we didn't take afternoon breaks. We stayed in the parks from open to close, and DD always took naps in her her stroller in the afternoon (and this is the kid who entirely gave up naps before her 3rd birthday). We also found that at night she might get tired and fall asleep in the stroller occasionally, and DH and I could keep going. DH and I would often take turns riding the big rides when she was asleep in the stroller.
I would take it with you and take it to the park- at least the first day. My plan would be to park it in an area, and then just do that area walking and then get the stroller, put her in and move over to the next area (or ADR or show or character meet you are trying to get to on time). You will still get plenty of walking and handholding time, but you might be surprised at how much she winds up using the stroller. When DD was 2 and 3, we used it a lot. When DD was 4 and 5, we might park it for a couple of hours and come back to it, but she always needed it at some point during the trip. Little legs can get really tired at big WDW!
To give you an idea, to walk entirely around World Showcase at EPCOT is approximately 1.2 miles, and that is only about 1/2 the park! You can walk MILES during a day at WDW.