A few years ago, a local community theater group here did the carpet ride by simply putting ornate rugs on top of 3' tall platforms on wagons, and having stagehands in
blackout garb push them about the
stage. With careful side lighting and followspotting to keep the hands out of sight, it was reasonably effective, and very cheap. However, it required a large
flat floor (no levels), and there was no rocking motion to the carpet, so no banking through turns.
Another approach might be a carpet
platform that doesn't move, but is able to be rocked on a center
point, and surround it with video or swirly lights to create the sensation of movement. If you're feeling brave, you can try a rocking top on a
wagon, but that might be more than the actors can balance with.
The Phantom's boat, in the professional version, is a radio controlled cart with
electric motors driving the wheels, probably beyond most HS budgets. A cheaper way might be to put the boat on a
wagon with swivel casters in front, and fixed casters in the rear, and tie two ropes to the front, one going to either side of the
stage. If the ropes stay low enough, the
fog can hide them as the kids pull the boat around on the
stage. You might need to affix a pulley to the floor offstage to keep the rope low enough, and keeping enough tension on the non-actively pulling rope will be essential to not running over a rope and getting tangled.
Some possible choreography: Start the boat USR (for instance), and use the SL rope to pull it diagonally towards DSL. Mid
stage, use the SR rope to turn the boat straight US/DS and sing a verse. Both ropes to travel straight DS, pull SL rope to turn and continue DSL, then SR starts pulling, and the boat turns and heads SR.
Like the simple carpet platforms above, this will require a
flat floor. Use the largest casters (and lightest actors) you can find, to make it as free-rolling as possible. Might need to draft some football players to be rope pullers... this is a cheap solution, not a sweat-free one.