Kevin, are you hoping for an overnight or day hike permit?
If you show up at the Visitor Center S of Lone Pine the day before you want to hike, chances are you can pick up a free walk-in permit. Week days are easier to get than weekends.
Check the Unused Whitney Permits
pages to see the numbers of what is left at the end of each day in the past. Those numbers are what is left at the end of the day, and NOT what is available when they are first made available. Your chances are very good.