I wouldn't. The Game Cube drive is not designed to be used in vertical position. It will cause the spindle bearing to wear out pretty quickly. It will also give you calibration problems over time since the lens-sled was not calibrated for vertical use.
I would think it would be just fine actually. The gamecube was very much designed to be portable unit, thus the whole way it is set up in the first place. Everything from the snap in disc holding, to the handle in the back.
When carrying it by the handle the unit is in a vertical position, and though it is not powered on and spinning, I would think the snap in design would be equal to that of a discman in funtion. They certainly can be played in any position.
Though I am no expert, and have not tried so myself, so... Take it for what it is worth.