Hi Bea!
Stroller comfort makes a big difference between a
happy Disney baby and a cranky Disney baby, so you get a gold star for thinking ahead, Mom! Your stroller will spend a lot of time in stroller parking through your Disney Days. (Strollers are not allowed to wait with you in attraction queues, nor are they allowed inside at Table-Service Dining Location.) So this will give your little ample opportunities to get out and stretch. However, there will be plenty of times when it's more convenient for your toddler to cruise around the 'World (the Walt Disney World) than walk, especially if your plan is for your babe to take stroller naps rather than go back to your Disney Resort Hotel to rest during the day.
I always bring our own stroller when we visit Walt Disney World Resort. It's was one less change in our daily routine, and I know my three-year-old son really likes his stroller. There are
strollers for rent in each of Disney's theme parks that steer really well, but are hard plastic and do not recline. So if you're thinking of renting a stroller from Disney then I would definitely recommend some kind of extra padding.
I do want to point out that the weather in January, the time of your visit, can fluctuate wildly. Some years its very pleasant, some years is very very cold. We visited in March where the daytime was absolutely wonderful but we were all very cold in the early morning and once the sun went down. Before our trip, I purchased a 'toddler size universal stroller footmuff' that was fleece-lined and kept my guy toasty warm during those cold hours. It also zipped on the sides like a sleeping bag so the top part didn't fall off like a blanket would. It provided a bit of extra padding but also a ton of warmth.
I hope this helped, Bea!
~Kristen