Hey there I have been on WDW busses with a 8 month old, 23 month old and then again 2 more times before she turned 3. Then I have been with a 7 month old and a group that inculded another 6 month old and 2 preschoolers.
We have never had any problems with the busses really. Once on the last trip my friends son was playing around with his aunt and the busses stoped and he feel forward hurting his head a little. He was fine by the time we got off the bus.
You will have to fold down the stroller and the kids sit on a seat or lap. They can not stay in the stroller unless it is strapped down like a whelle chair.
I never had to stand with the baby, someone always popped up and gave me a seat the second I walked on. My friend did stand a few times, not sure why I always have better luck then her with people. Hubby had to stand one night with little PhD who was asleep and he was supporting the stroller too.
If you are really concerned you might want to consider a rental car.
I cut and paste this from my last trip report. It was rather flabergasting:
I just remembered something some people here might find interesting (mostly the stroller obsessed mom's.)
One day a couple gets on the bus. The father goes to the front of the bus and explains to the driver that he needs to tip the bus to get their stroller on, it would not fold up. The driver explains to him then he is not able to take it on the bus. They go back and forth a while and then magically the driver decides to let them on. So she loads this double stroller (the Peg Pergo with "boots" a front back twin, the one that goes for around $500 +) on the stroller, with the seats still on it and the kids still in the seats.) The couple with 2 kids around 1 year old each gets in. In addition the stroller was loaded down with tons of stuff in the basket under it. The driver then makes the people sitting in the HC seats move and spends 10 min locking down stroller the with the Handy Cap straps. The man was the nicest man ever, the women looked rather ed (at the world in general.) My curiosity got the better of me, I had to ask them, "If it doesn't fold down, how in the world did you get it to Disney World." The father explained that oh the seats do come off. Then he looks at my Peg Pergo Aira Twin and say's "That is a nice stroller, but we thought we might have trouble getting it through doors."
Can you imagine going through that every time you get on the bus? The kids looked perfectly normal (ie this was not a stroller/wheelchair.) The mother did get one out and sit him on her lap. She didn't make eye contact with anyone or talk on the bus ride. Boy that poor man was whipped!
Well in all my years staying at WDW property and using the buses I have never seen anything like it before. I was glad we didn't encounter it again on the trip. I don' t mind waiting for people in wheelchairs to load and be strapped down in the least bit, I am happy they are out enjoying WDW too. But hey, I had a double stroller and two kids too. (not to mention my Peg Pergo cost over 200 less then that one!) So when people ask for reconditions for a double stroller, that isn't it!