Gosh, that's frustrating. Is your child losing weight too? Have you ruled out things like stress? I know that sounds crazy but is a pretty classic sign for teens that have anxiety. I would have your child get a full physical. FULL, as in everything. Continue working with your GI doc. I had cyclic vomiting as a child myself and it was due to a duodenal ulcer. It was found after about 4 months a thirty pound weight loss. But once you've gone through all the evaluation that you could, there is also this to consider. https://www.webmd.com/digestive-disorders/what-is-cyclic-vomiting-syndrome#1 This cyclic vomiting syndrome is rare though but does happen more in kids than other populations. Antacids may help and make sure your child stays hydrated.