The sentence says the Robot "May be controlled remotely by radio, infrared, or wired control Boxes to the robot". LEGO MINDSTORMS Robots are programmed and not remotely controlled. Would this type of robot be acceptable?

Yes. Programmable controllers on the robot or in a remote box are allowed. But if the robot is touched after the Event Supervisor says "1, 2, 3, go" the time will stop per rule 5.f.iv.