A question about gag reflex If left glossopharyngeal nerve and right vagus nerve are intact, but there is a lesion at left vagus nerve, would stimulating the left side of pharynx elicit a gag reflex?

The sensory and motor components are required (GP and vagus respectively). Similar to the corneal reflex and cranial nerve 5 and 7, if a lesion exists in either the sensory or motor component there will be a defect. In your case, there is no motor innervation on the left due to the lack of the vagus nerve so there is no vagus nerve response.