Sara K, your friend should prepare and go on your trip. July 7 is a Tuesday, a good day to pick up unused permits.
If nobody here offers an open slot, here is what you should do:
Keep watching the recreation.gov site (here's
a link for your week). Bookmark it and check every day. Especially in the last two weeks before your trip, a few slots may show up since a few really nice people will log in and drop their unneeded reservations.
If that doesn't work:
Look at the
2014 unused Whitney permits page. You can see that on the Tuesday after July 4, 13 out of 60 overnight permits went unused. (It's an anomaly that on Wed, there were none. But look at prior years and get the averages.) (By the way, you only need the permit for ONE day -- that one day permit is your permission to stay in the Whitney Zone for up to 2 weeks! The important thing is that you START your hike on the date shown on the permit.)
If you go in on Monday, July 6 and pick up your two reserved permits, it is VERY likely that there will be plenty of others available, from people dropping out of group trips. When the group leader picks up the trip permit, those unused slots become available. The numbers you see on the Unused pages are what is left AFTER people walked in and requested unused slots. You are sure to get one of the slots.
On Tues, July 7, at 11 AM, all the complete no-shows are made available for the July 7 entry date, too. So even more permits will likely come available then.
