Could a hairline fracture in the radius cause stiffness and make my thumb pop when moved?

Not likely: Sounds like an unrelated problem.

One associated: issue that arises occasionally with a fracture of the radius ( and for some reason nondisplaced fractures) is irritation of the extensor tendon that elevates the thumb tip or the EPL tendon. Another cause of "popping" can be a separate issue in the thumb itself such as a trigger thumb with irritation from the injury or swelling due to it. These issues should be discussed and examined.