Well, like everything in the celiac world, this is inexplicably complex.
Vinegars made from rice or red wine or apples are absolutely, naturally gluten-free. Of course, as with everything else, check your sources to make sure the vinegars are pure and not cut with anything else. But apple cider vinegar, red wine vinegar, white wine vinegar, and rice wine vinegar are clearly gluten-free.
Distilled white vinegar is made initially from grains, sometimes with wheat. It’s clear from the myriad of tests and every celiac association there is that the distillation process removes any trace of gluten. These vinegars always test at less than 20 ppm. They are considered gluten-free. I have no problem with distilled white vinegar and I am very sensitive to gluten, immediately. However, some small percentage of celiacs report having gluten-like symptoms with distilled vinegars. Check out this guide by the Celiac Support Association for more nuance on this.
(By the way, the same is true for distilled alcohols, such as whiskey, vodka, and gin. Even if they began as grains, they are gluten-free by the final product. Good friends on Vashon started a distillery called Seattle Distilling Company and their whiskey, vodka, and gin are made with as many local ingredients as possible. They’re also delicious and gluten-free.)
Malt vinegar contains barley and thus is not gluten-free. (However, the Coeliac UK association says malt vinegar is almost free of barley from the fermentation process.)
So, to sum up: yes to apple cider vinegar, balsamic vinegar, red and white wine vinegar, sherry vinegar, and rice wine vinegar. Yes to almost everyone for distilled white vinegar. No to malt vinegar (well, maybe).
There’s one thing about having celiac: it’s never boring.