Many people ask me about supplements for their puppies. Theoretically dog food contains all of the balanced nutrition your dog needs. If it's dry kibble, it's processed and dried. Any dried food preserves more than canned or frozen, so that's good. But depending on their process, it still isn't a raw diet or fresh. It seems to me that most of my dogs do just fine on high-quality dried kibble food. But there are certain times and certain ones that seem to need a little extra something now and then. I supplement with fish oil first if there seems to be a skin or coat issue. (I do this for my children too!) I took a picture of one of the ones I use because it's a middle-of-the-road example.
There are high-quality fish oils like Carlson's for human consumption (cod liver oil.) This particular brand is available at Wal-Mart for around $10 and is Salmon Oil. Here in Boise, Idaho where it is a high desert climate with low humidity, it doesn't hurt to have a little extra oil for our skin and our dog's skin. There are other supplements I use for my breeding dogs, but they usually aren't necessary for dogs that are not breeding. I recommend asking your vet what "vitamins" they recommend, and there are many out there that would like to sell to you! Watch your puppy closely for any deficiencies and if you notice anything, it's good to make a change slowly.