Everyone seems to use radio control, I have wondered if a wifi link to a Pi3 or Arduino would give a more flexible control system.
Also you can buy rechargeable drills with variable speed trigger quite cheaply, I wonder if that would make a decent starting point for motive power giving you motor, gearbox and speed controller.
Something withing the budget of a secondary school class/club could be pretty cool. The school my kids go to is already well equipped with CAD systems, 3d printers and cnc machines.
And if the bot goes all skynet, the house robots can just end it