Amie, what John said. It depends on the dates you're looking at, but if you are flexible enough to start mid-week you shouldn't have a problem except possibly around a major holiday. I've been traveling out there almost annually since 2009 from Atlanta to climb Whitney, usually with my family. The first year I had a permit from the lottery, and I had the same concern you do - long distance, tight itinerary, can't possibly take the chance of hoping to snag cancelled or no-show permits. MUST have permit in hand before committing to the trip. I've learned better.
Every trip since 2010 has been with the full expectation I'd be able to snap up 4-5 cancelled or no-show overnight permits for a mid-week start. Batting a thousand so far. I completely ignore the lottery, knowing that as long as I avoid a Fri or Sat start, or holiday weeks, I'll get my permits. It's amazing how many permits go unclaimed mid-week, even in the height of summer. People from all over the world get a lottery date in March, but stuff happens and they can't make it. Few are considerate enough to cancel and free the permits up since there's no refund, but some will cancel a few days before the permit date. You can reserve these by calling the Bishop Inyo NFS office (the website availability is not always current), but you'll pay a reservation fee. The other option, for an overnighter, is to show up at the Lone Pine InterAgency Center the day before you want to start and claim available permits for free.
Worst case scenario: somehow, someway, despite all the history, no Whitney overnight permits are available. You divert from the Whitney Portal Road up to Horseshoe Meadows and do an overnight trip to Mt. Langley, another fourteener just south of Whitney. Permit required there as well, but a tiny fraction of the demand Whitney has.
