We rode it in late March and didn't see a sign, but we got on in the handicapped area (you still wait in the regular line, but you board at a place where your "boat" is trapped and kept still for boarding). The other people on our boat took their shoes off and put them in the middle, but they did that after we were actually started up the first hill!
There is a rail all the way around the center of the boat for you to put your feet on to keep them out of the water. My youngest DD was not able to do that and her shoes got completely soaked. The rest of us (me, DH, my mom and my ten yr old niece) kept our feet up on the rail and no one got their shoes wet (although my mom got completely soaked otherwise).
I wouldn't be surprized if they require shoes on the ride. The other people in our boat had to scramble to get their shoes on when the ride was finished (and 2 of them got off holding their shoes).