The same thing happened to us on our trip in 2007. We were taking a bus from the Yacht Club to the Magic Kingdom to transfer to a launch to catch Hoop Dee Doo at Fort Wilderness. I was sitting very near the front of the bus and I was watching out the front window.
As our bus was traveling north on World Dr. getting close to the MK I could see the road up ahead. There was a line of traffic crossing the road in front of us, barely stopping at the posted stop sign. As our bus got closer to the intersection (we're weren't supposed to stop) I was counting the cars rolling through the stop sign. I had an uneasy feeling that there was going to be an incident judging by the aggressiveness of the drivers of those cars, it was like they were all on their way to a fire or something. Just as our bus entered the intersection an idiot in a red car blew the stop sign and crossed directly in front of us, I call the guy an idiot because anyone smart driving a compact car would never play chicken with a bus bearing down on them going 40 miles an hour. Our bus driver must have anticipated this happening because he had slowed down a bit before. Our driver slammed on the brakes hard, all the passengers were shoved forward. Luckily there wasn't anyone standing at the time our bus wasn't too full.
A couple of small children, including a baby rolled off their parents lap and fell to the floor of the bus. One lady banged her head on a steel rail attached to her seat. She had been dozing off at the time, she said later.
Luckily we avoided a collision and the red car sped off and never stopped, the angle of my line of sight was poor and I was unable to get a plate number of the car.
Our bus driver continued on he asked if everyone was ok, some people said yes some said no, kids were crying, the bus driver looked shaken up. The bus driver called in the incident on his radio, by this time we were turning into the MK bus area. Instead of parking at the YC/BC slot we continued on to a more deserted area and parked. Several Disney management and security types swarmed our bus. As the bus driver was being debriefed security people were going to each passenger asking how they were. My family and I were fine, as were most everyone on the bus. Only the lady who bumped her head wasn't doing so good, but she turned out to fine as well.
I was able to speak with the bus driver a little, a young guy in his 20's. he said he was new, and had been driving the bus for less than 6 months. He was worried about getting fired. I told him I saw the whole thing as it happened and there was nothing more he could have done, it would be BS to get fired for something that wasn't his fault. Yeah, a few passengers got knocked around a bit but nothing serious, he avoided a collision, which I assumed wasn't easy to do, handling a big machine like that.
Someone, I assume was the bus drivers supervisor overheard our conversation and asked me if I would fill out an incident report giving my account of what I saw. I was at first worried about making it on time to our dinner show reservations, but we still had time, so I abliged.
I gave them my personal information, cell phone, address and such and I told the driver in front of his boss that I appreciated all that he did.
A different manager offered to drive us in her car to Fort Wilderness, but I declined and said we would be happy to take the launch. So she at least escorted us to the gangway. She gave our kids some stickers and we were off.