Nighttime-only reflux is a pretty common thing for people with GERD. A lot only experience their symptoms at night, with some of the symptoms being coughing and sore throat. (Haven't heard about vomiting, though that doesn't exclude it by any means.) Has he tried lifting the head of the bed a few inches (6-12) and not eating 3-4 hours before bedtime? Try that out and see how it works. Of course, also follow CalGal's advice.
There is another possibility, and this goes with the whole "side effect of the surgery" train of thought: the mesh may be coming undone. (I presume that by "GERD surgery you mean a Nissen fundoplication. Partial or whole?) This is something the doc would have to check out.
As for the acid blocker, maybe he's not reacting well to that particular one. Ask for a different prescription. (Guessing that they started him on an H2-blocker like Zantac. He might need something stronger, a PPI like Aciphex or Prilosec.)
The Vagus nerve problem you're talking about is probably gastroparesis, and while that is a possibility, unless he's throwing up undigested food in the morning, or hours after eating, I wouldn't worry too much about that just now.
Any idea why he only gets sick in the morning?
If he's bringing any form of blood up when he gags and/or throws up, he needs to contact his GI doc and get it checked out. He may also need to be re-evaluated via pH monitoring in some fashion to find out whether or not he's refluxing acid and if that could be the cause of the cough he's experiencing. If that's happening, he may need to consider changing his diet, if necessary, and possibly going back on some form of acid suppression medication.