That's a tricky question, it depends on the power supply and the LED outputs. What voltage does this thing run off?
There are several solutions.
You could replace each LED with an optoisolator and use it to drive a transistor or relay to control the 12V lights.
You could build a driver circuit with transistor to buffer the output from it so it can drive 12V lights.
You mighe be able to run this from 12V, remove the LEDs and current limiting resistors and replace them with low current 12V lights, but this is a very risky idea which could ruin the Quizzard.
The chances are you'll need to reverse engineer the Quizzard to some degree so you know the nature of the LED outputs, i.e. whether the LEDs are powred from a current source or sink.
Why have you decided to do this? Didn't the other circuits work? It's probably easier to get them working rather than reverse engineering and modifying an existing product.