I also say bring your own stroller! We travelled to WDW twice since we have had kids. The first time we had an almost 3 yr old and a 7 mos old and we brought strollers for both. The next year we brought just one for the 20 mos old and rented for our older dd. We ended up investing in an $80 Graco stroller that would recline, had a basket and had a sun visor. It wasn't as big as our regular Graco one.
The WDW strollers are ok for older kids (like 3 and up), but if your child wants to nap and recline they can't do it in those big plastic blue ones comfortably (same goes for a cheap umbrella stroller). You at least need a stroller that can recline for naps, has a basket underneath for your diapers and "stuff", and has something on the top for some sun protection.
Plus if you go anywhere else, you have your stroller - the mall, DTD, walking in the airport right up to the gate, evening walks, etc. If it is very hot, it will be even hotter if you are carrying a baby everywhere you go. That can sure make for a cranky baby

Plus travelling around WDW in their stroller they can see everything ... it is so entertaining for them!!
This year we are travelling with an almost 5 yr old and a 2.5 yr old. We are talking a stroller for the younger one again and will rent for the older dd. We will need the stroller to contain our youngest in the airport this time!!!
Good luck with whatever you decide to do!!
Kerri