Can intubation cause damage to the vocal chords?

Yes: Endotracheal intubation involves placing a breathing tube into your mouth and passing it between your vocal cords. It is not uncommon to have hoarseness and a sore throat that lasts 24-48 hours. Longer term injuries are possible but unusual.

Possible, unlikely: The "typical" endotracheal tube, made of flexible plastic, is usually slid in between the vcs under direct vision. If there is difficulty observing the insertion, the tube may be pushed against the vc causing irritation or - rarely - more sever injury. The presence of the tube laying against the vc for days in the ICU can cause irritation or abrasion. Sore throat is also caused by dry oxygen.