What could cause sudden onset of joint stiffness and pain in the feet?

Arthritis. The most usual cause of pain and stiffness in a foot is degenerative joint disease, or loss of the joint cartilage. In the early stage, this may not be noticeable unless you have walked or run much further than usual. As the condition progresses, it may begin to hurt more frequently and adjustments should be made to shoes for extra cushioning and support.
Many Things. Gout, nerve entrapment, trauma, tedonitis more info on the symptoms would help with better answer.