Does the vagus nerve play a role in the onset of anxiety or depression?

I don't think so: The vagus nerve is a mixed cranial nerve with both sensory and motor fibers. The main function of the vagus is autonomic, it also provides motor innervation to the palate and pharynx and supplies sensory and motor input to the larynx and vocal cords.

Not that I have ever: Heard. However, the vagal nerve can be stimulated to help alleviate depression. The implanted vagal nerve stimulator is one form of treatment, neurostim treatment is also available although not fda approved, and acupuncture which stimulates the ear lobe can all help alleviate depression.