Hi Gina.
Early February is one of my favourite times to visit Walt Disney World with my family. Great, mild weather and shorter lines make us all happy!
If you're fairly confident that you'll only need a stroller for your 2-year-old & you travel lightly in the parks without many bags, you may find that a light-weight single stroller from home is convenient for you & all you really need.
However, if your 6 year old tires easily & you want a "back up plan" for days where he/she may be tired from the previous day's activities - the double stroller is very handy.
Walt Disney World's rental strollers are lightweight to push, manoeuver easily and provide great storage.
You can transfer your rental from park to park in the same day - picking up a new one after entering the next park (keeping in mind that stroller types are subject to availability) if your ticket allows "hopping." There is also a $2/day discount for single strollers and a $4/day
discount for double strollers if you pre-purchase a multiple day usage
of the stroller(s).
One of the biggest perks I found with the double stroller when my older child was 'almost' past the age of requiring one, was that it was a great 'storage' vehicle while in the parks. The canopy is sturdy enough to place some lightweight bags on top and the pouches on the back of the canopy are so handy for easy access to snacks, sunglasses, sun lotions etc. When waiting for a parade or fireworks show to begin - you'll also have a handy, extra seat for someone with tired legs to perch on!
What worked really well for me when my children were 3 & 6, was to rent a single stroller one day - and a double the next (alternating through the trip). That way, my 6 year old was able to rest her legs for a couple of days during our trip.
Have a Magical time on your vacation!
Kirsten