Permits for entering most trails in Inyo National Forest come available online through the recreation.gov website exactly 6 months before your hike start date. If you want to be sure to get one, plan early and get one on that day. The entry permits are by specific trail heads, so decide which one you want.
If you want to exit via Mt Whitney -- Trail Crest down to Whitney Portal, that requires a special "Trail Crest Exit Permit" Those are difficult to get, and you have to get it at the same time you reserve your entry permit. Since they are in such demand, getting one the first day, 7 AM is best. For your exit, you need to get the Permit Type of "Overnight Exiting Mt Whitney". Just checking the site, you can book May start dates now (6 months ahead). The available dates will roll forward, not sure if it is daily or less frequently.
You can summit Mt Whitney even if you don't have the Trail Crest Exit permit. You could climb the trail from the west, then descend again to the west and take another exit. All other exits are open -- no exit permit required.
