At Whitney Portal, across the street and across the parking lot from the trailhead, there is a small walk-in campground just for backpackers. This place is pretty cramped. I stayed there once and never wanted to stay there again. It is better than nothing.
A half-mile below there is the Whitney Portal Family Campground, so it is full of RVs and kids. In prime season, this place is reserved well in advance. On some years, I have had trouble booking a spot there three months in advance.
With this in mind, some backpackers avoid the overnight stay at Whitney Portal by camping their last night up at Lone Pine Lake. It is about 2.4 miles up the trail from the bottom. Then they walk that last hour down to the trailhead, but that is likely to get them there in the morning, and there aren't so many cars going downhill to Lone Pine at that time.
–B.G.–