Danielle, two words:
Unused Permits If you have several days available to hike, I would bet you would get walk-in permits the first day you tried.
If you don't get them the first day by 11 AM, head up to Horseshoe Meadows and hike to Trail Peak or Cottonwood lakes. Then head down to the Visitor Center and try for permits again the next morning. You CAN get permits.
Also, check online on May 1. There will be a few released "at random times". They are the lottery-won slots that people neither confirmed nor declined.
Finally, start checking recreation.gov two weeks before your planned trip. A few good people will login and drop their slots that way, too.
