The Short Answer:
Chinchillas can eat grapes, however since grapes are high in sugars, they should not be given often or in large amounts. Many chinchilla owners will give their chinchilla grapes as treats on occasion, which is perfectly fine.
Giving your chinchilla a large amount of grapes can be a dangerous thing, as it can lead to obesity, and diseases like diabetes. In the wild, chinchillas won’t come across grapes often, if at all. Their bodies are not made to process high sugar foods.
The First Time:
Like other animals, and humans, some chinchillas can be allergic to grapes. Although this is somewhat rare, it can be deadly. The first time you give your chinchilla a grape you should watch it as it eats it, and for a while after to make sure it is ok.