If you can find a motorised up/down mechanism it should in theory be possible to replace the motor with a hand crank or something like that. But it might be just as easy to find a way to power it in place. Car seats will work with automotive electrical power which would usually be 12V DC or maybe 24V. Not sure what kind of amps it would draw but it should certainly be possible to power it from a PSU you could plug into the wall, or even a car battery you could take to the events. Provided it’s isolated in a closed container that needn’t pose any risk to the kids attending.
Depending on how far you want to go in terms of self-build, you could potentially create a seat base out of something like a motorcycle scissor lift and mount the seat on top of that (with a surround to stop curious fingers and legs from getting in places they shouldn’t) or you could do something with the gas lift from an office chair, or linear actuators (though these would require powering). All are designed to cope with much more weight than a regular person.
You could have a seat which is static except that it has, say, actuators on each leg which can extend to raise the height. Build it to the lowest height you want (for the tallest occupant) and then use the actuators to lift it up for smaller occupants. Or you could make extensible legs using just a slide-and-pin system which you could adjust manually and would not require power or hydraulics.
For back and forth motion, as mentioned above, car seat rails do a good job. I have those on my cockpit seat which is not height-adjustable but does move in and out for different leg lengths on the rudder pedals. Or you could have the pedals themselves move back and forward.