I don't think you'd want to use servos to drive the wheels for something like this. You probably want very fast steppers (so you can get precise rotations) or geared DC motors (so you can move very fast). I'm assuming the robot has some very basic black and white computer vision, or maybe even simple laser range finders, which would be simpler to work with and more likely to generalize to new mazes.

Here is an example of an inexpensive laser range sensor: https://www.ebay.com/itm/TOF10120-ToF-Laser-Range-Sensor-Las...