Fishing is allowed in any of the lakes inside Sequoia and Kings Canyon [1]. You can purchase a fishing license and tackle at the markets at the Grant Grove, Cedar Grove, and Lodgepole visitors centers.
The visitor centers can also advise about season opening and closing dates. Anglers are permitted up to five trout per day. The lakes and the Kaweah drainage hold rainbows, eastern brook, German brown, and golden trout.
Links:
[1] http://www.moon.com/destinations/california/yosemite-and-the-eastern-sierra/sequoia-and-kings-canyon-national-parks