Why is it that my thumbs keep locking or clicking?

Trigger finger: Typically has a locking or clicking tendon, other things can click in a finger or thumb such as a joint, a tendon snapping over a bone spur , other areas of tendon irritation in the wrist, even a cyst or tumor on a tendon or just a joint click from. Normal motion but yes it sounds like the description you give is that of a trigger thumb. Many get permanent relief from corticosteroid injection.