Your chances of getting a day hike permit for Sunday are quite good if you show up at 2 PM Saturday. If there are more walk-in requests than there are available no-show permits, you draw numbers -- a mini lottery.
But so far this year, they have not run out of no-show permits for anyone walking in. Sundays would be the first to "sell out", and expect that to happen the July 4 weekend.
You can view the walk-in permit options in the "Wilderness Permit Options" link to the left, specifically, read box 6,
No Fee Permit Options.You can see the unused permit numbers here:
Unused Whitney Permits - 2009