retrofan, if you go and cannot find a first-come, first-served site, consider the "ravine campground", a tucked-away walk-in set of sites, downhill from the overflow parking lot. The regular backpacker walk-in sites are closed due to the recent earthquake and rockfall damage, and I'm not sure if it will reopen this season. So the ravine campground may see a lot more use. But it's worth checking out.
Here's a link: Sleeping in your car at Whitney Portal
This map/picture is shown in that thread: