The Short Answer:
Chinchillas, like humans, can enjoy eating apples. Apples, however should be given in moderation, and only as a treat, as they are high in sugars. In the wild chinchillas would not have access to apples, so their digestive system will not be used to them. Giving a large amount could unintentionally harm your pet.
Apples are high in sugars, and high in vitamin C and Fiber, making them a healthy snack, or treat. You should not, however, have apples as one of the main food sources for your chinchilla. Giving your chinchilla a few slices of apple a week will be ok, but do not give apples as a daily routine.
The First Time:
Like all other animals, it is possible for your chinchilla to have an allergy to apples. Although it is rare, if your chinchilla has an allergic reaction it can lead to death. To avoid any negative effects, first give your chinchilla a small amount of apple, and watch him or her for around 15 minutes. If he or she does not start acting oddly, then he or she will likely be ok with having more. If in doubt, make sure you ask your vet about giving apples as a treat for your chinchilla.