Can you damage your shoulder from whiplash and not feel it right away?

Possibly, yes: The most common shoulder pains associated with whiplash are actually referred pain caused by inflammation or compression in the cervical spine around the nerves that innervate the shoulder region; it is also possible that the shoulder was injured independently at te time of the accident and the whiplash was a distracting injury. You should seek evaluation by a physician if your symptoms continue.

Days: Any injury from a motor vehicle accident, including whiplash, would present itself within days or a week. It would be unlikely that months or years later you develop a condition that was caused by whiplash.