I was there twice when I was 9 years old, Flushing Meadows in Queens, I believe we went through IASM sitting in a '64 Ford Fairlane convertable or maybe it was a T-Bird. I saw it again at Disney World in early 1973 when I was a senior in high school. Orlando was still a small sleepy city, the airport was still called McCoy AFB and there was nothing but orange groves, pine trees and gator ditches between Orlando and Disney World.
I too visited the Worlds Fair that year and was also age 9 the first year (for those that don't know, the NY Worlds Fair ran the summer of 64 and 65). My dad was an executive that dealt with almost all of the corporate sponsors and we got VIP tickets for most of the rides. For Carousel of Progress that included walking on a catwalk over the animatronnic characters on their stages. Did you know that they didn't bother making the parts of the bodies the audience couldn't see? Very disturbing to a 9 year old.
I think you're combining two rides in your memory. I'm pretty sure Small World, sponsored by Pepsi, always used boats. Either Ford or GM was right across the 'street' and it used cars. That's the ride that had the animatronic dinos, I think.