Rain and a puppy is a great thing and the pup won't get sick. With my puppies, every time it rains, I go out with them and as soon as I step outside I start to play with the dog like crazy and I get all happy and excited. This way the pup learns early that rain is a good thing and nothing to be scared of. it comes in handy - trust me. So get out their and play with the dog in the rain!!!! I would not leave the dog outside in the rain with no shelter for a long time, but playing is a good thing. The more you expose the dog to during this 8-16 week range the more adjusted your dog will be. Anytime the pup is going to experience something new, have some treats ready. Dogs do not get colds like we do.
PS - A downpour is a different story. If you can barely keep your eyes open out there, your dog won't like it much. But a steady rain is good.