I would be highly skeptical of this claim. The mechanism that this is supposed to work by is as some sort of antibacterial agent. Although, the mild acid in vinegar really is nothing compared to the acids that are already in your stomach. The main reason that vinegar is associated as a food poisoning remedy could be because vinegar is often used as a method of preserving food for longer periods (i.e. pickling).
Or as is mentioned at the Washington State University site
ORGANIC ACIDS: As you recall, all microbes require an #optimum pH or acidity in their environment to grow. If there is too much acid or base, a microbe will not grow. As the by-products of many microbial fermentations include the production of chemicals like ACETIC ACID (vinegar), LACTIC ACID, and PROPIONIC ACID it is not too surprising to find that humans, and other life, can actually use these substances as nutrients. However, when they are added to foods in sufficient quantities to lower the pH below that which will support the growth of most food-spoilage microbes, they can serve as natural food preservatives. Again, our ancestors recognized that "SPOILED" foods such as milk and certain vegetables, retained their nutrition upon becoming acidic and remained eatable (preserved) for long periods. Thus was born choice food items like yogurt, sauerkraut, pickles, cheese and buttermilk. Artificial acids, like benzoic acid, inhibit the growth of some molds, thus it is added to breads and other bakery products that require long shelf live. In many foods, like the sauerkraut you made in lab, salt is combined with acids to preserve food.
This site from the University of Maryland Medical Center states
Apple cider vinegar is a traditional remedy that, although it has not been studied scientifically, may have some antimicrobial properties. Mix 2 tsp. in one cup warm water and drink several times a day.
BOTTOM LINE: No studies back it up, but there appears to be no harm in doing it as long as you dilute the vinegar.
Keep in mind, if someone has ingested a non-food poison, Poison Control Centers specifically state
Do not give salt, vinegar, citrus juice or induce vomiting unless instructed to do so by the Poison Control Center.