UPDATE: I talked to the people at Inyo National Forest today. They told me that they're having a meeting tomorrow (May 20) to discuss when they'll start issuing wilderness permits again. This is apparently the hot item that they know they need to discuss.
Anyways, I would expect another press release later this week that explains when and how wilderness permits will be made available. I was told that the "how" part might include online-only registration.
From the Inyo Facebook page: https://www.facebook.com/inyonf/
"Inyo National Forest Provides Wilderness Permits Virtually, Effective June 1, except Mt. Whitney
Starting on June 1, the Inyo National Forest will be providing wilderness permits through a virtual process.
People who have reserved permits can call up to 14 days in advance to have their permit emailed or printed and left in a drop box at the selected visitor center.
The remaining “walk up” permits will be reserved through www.recreation.gov
starting up to 14 days in advance of the date of a wilderness trip.
The Eastern Sierra Interpretive Association staff will sell maps, rent bear canisters, and other products under a modified operation. Visitors are highly encouraged to purchase products online in advance of their visit at https://sierraforever.org/.
Visitors who reserve a permit will need to check by phone to receive their permit and the wilderness orientation, which will include:
Food storage/ bear can use
Human waste disposal
Pack it in, pack it out/ Leave No Trace
Specific guidance if their itinerary enters Sequoia and Kings Canyon National Parks or Yosemite National Park.The exception to this is Mt. Whitney day and overnight permits, which are being cancelled one week in advance of the hiking start day. Please see Inyo County guidance. Mt. Whitney is a high use trail and where rescues and recoveries are common. Whitney Portal Road remains closed at the lower gate.
People who have been awarded a permit reservation for Mt. Whitney on the Inyo National Forest will receive a notice from recreation.gov that notifies them that their reservation has been cancelled and a refund have been issued. There is no action required from you at this time. The credit card used to make the original reservation will be refunded."
Based on the bolded text above, I'm guessing that Whitney is going to remain closed until the switchbacks are usable (in around 4 weeks?).