Would speech therapy help a shy child with selective mutism?

No: Selective mutism is a manifestation of chronic pattern of anxiety. Drug therapy to relieve anxiety plus psychotherapy would be appropriate for your child.

As above, and . . . : Selective mutism is a sign of anxiety disorder - a form of agoraphobia. But a well trained speech therapist could be a helpful participant in the overall care of this type of patient, coordinating with psychologist and physician.