If sick after eating peach yogurt, do I have a peach allergy?

Not necessarily. Yogurt contains a variety of ingredients and is a milk product. If by sick you mean upset stomach, bloating, etc, you might be lactose intolerant. See if some lactaid helps with your symptoms. If you don't get these symptoms when eating a regular peach, i wouldn't suspect the peach in the yogurt. Might also check what kind of additives/preservatives are in the yogurt.