Do you mean this stroller?
http://www.strollers.com/asp/show_detail.asp?sku=PGR1199&refid=FR67-PGR1199
I have a story about that stroller.
I once saw a couple get on a WDW bus with this stroller. (or at least one that looked like it, it was maybe 2 years ago.)
They argued with the driver for 5 min about how it would not fold down, and insisted the driver "tip the bus" like for a wheel chair. They then made people who were sitting in the handy cap seats get up so they could strap the stroller in and then take the kids out (not handi cap kids) and hold them to sit in seats on a mostly busy bus. If was an afternoon. The kids looked around a year old. It did take a least 10 min longer then it should of.
I have no problem at all waiting for a person in need of a wheel chair while on a bus. But a stroller?
I asked the father why they didn't fold it up. He said it would not. I then asked "how did you get it to the resort?" He did then hem and haw and say you could take it apart, but it wasn't easy. The whole bus was glaring at them. The father tried to be friendly, the mother just sat there looking, well on Disney like. I felt bad for the dad.
Then he looked at my peg pergo aira twin and said "we thought about that one, but it looked too hard to get through doors."

That about floored me because it folds up small and weights 14 lbs. It is rather easy as far as doubles on WDW busses go. It is the same size a wheel chair so it fits through doors fine, as long as you "hit them right."
I forget to add,
I think it should be fine on the monorail. They often direct us to the HC car and ours fits fine there. If someone else needs it, (who really is HC) we wait for the next one or fold down. So far it has never been an issue.
If it doesn't fold up you might have an issue on the trains. You will just have to leave it and come back for it, like the rented strollers at the parks. Other then that you should be fine in the parks.