Alabama Hills are close by but can be pretty hot. And dry. There are camp sites outside the Portal on FS land ($15-20 a night). Creekside. And there is a backpackers camp right at the Portal.
The Alabama Hills have very nice car camping that is pretty close to town, some fun sport climbing and good views. It doesn't have much water though. It tends to be hot, but the rocks can provide good shade in the afternoons if you pick your site well.