People have taken 6-year-olds to Whitney!

Make SURE you spend some time at altitude before starting the hike. Mammoth is at 8k elevation, and is a good place to start. Horseshoe Meadows is at 10k, and is even better, so long as the kid doesn't get AMS there.

You should sleep at the 8k or 10k elevations at least two nights before hiking up the trail.