Can degeneration of the C5-C6 vertebra cause episodic syncope due to vagus nerve damage or entrapment?

No but VBI might. VertebroBasilar Insufficiency (VBI) - The vertebral arteries run up the back of the neck thru tunnels (foramen) in the vertebrae (bones of spine). Some neck maneuvers can compress the artery where it passes thru the foramen. While not usually syncope, this can cause vertigo & drop attacks via buckling of the knees usually in older folks with degenerative spinal problems. aka beauty parlor syndrome.
No it can not. The vagus nerve is the longest cranial nerve. It contains motor and sensory fibers and, because it passes through the neck and thorax to the abdomen, has the widest distribution in the body. However, it does not run through the cervical spine and thus would not be affected by vertebral pathology at C5-6. There are many causes for syncope and should be investigated with your doctor. Take care.
Not related at all. The vagus nerve as it may relate to syncope when you mention some type of damage or entrapment is anatomically unrelated to anything having to do with C5/C6 degeneration or the spinal cord/spinal nerves in that vicinity. The right and left vagus nerves enter the body by coming down from the cranial point of origin through the neck and into the thorax well in front of the spinal cord, or C5/C6.
NO. NO nerve entrapment in this area causes cervical radiculitis and more uncommonly motor difficulties in the arm. Hope this helps Dr Z.