A permit is required all year round to day hike and overnight camp to the summit of Mount Whitney. Quota season is in place from May 1 – Nov 1 and you can get your permits through the Mt. Whitney Lotto.
You can, however, hike the first 3 miles of the Mt. Whitney trail to Lone Pine Lake without a permit which is a beautiful hike on its own. Some people use this as a training hike the day before their Mt Whitney attempt. Once you reach the Whitney Zone just beyond Lone Pine Lake, a Mt. Whitney permit is required and rangers do check.