I'll just jump right out there and encourage you to bring your own. I find that the times we needed a stroller the most were walking out from the parks and getting back to the room. Children, especially the little ones, tend to fall asleep the minute they get on Disney transportation. Being able to place them into a stroller without waking them up to get back to the room is priceless.
And since you are staying at the Polynesian, you will be able to take the monorail to both Epcot and the Magic Kingdom. Most guests don't fold up their strollers on the monorail, so that is even better for tired princes and princesses.
The cost of renting strollers has also increased quite a bit this spring (2008), so much that if you rent a stroller each day of a trip you could probably purchase a brand new stroller. Here is a
link to the cost.
If you would still like a rent a stroller, they are a great design (no padding, however) that push easily. My sons always enjoyed riding in them and had no complaints. Also, I did not notice any lengthy lines for strollers on our recent trip.