Does a guitar with active pickups and stereo output jack require cable for the to function?

A guitar with active pickups and a stereo output jack does not require a special cable to function. A regular guitar cable with mono 1/4-inch plugs will work just fine.

The stereo output jack is simply there to provide the option of sending a separate signal from each of the guitar's pickups. This can be useful for creating stereo effects or processing the two pickup signals differently. However, if you're not using this feature, you can simply plug a mono cable into the stereo jack and it will work just like a regular guitar cable.