If your head is still reeling from the considerations relative to providing a greener diet for your puppy, you'll be glad to know that making greener grooming decisions is a bit less complicated.
Think about the grooming supplies you may need for your puppy, then think about the green motto of reduce, reuse, and recycle. There are the essentials: brushes, combs, toenail clippers, and a toothbrush and tooth-paste (and possibly other items depending on the kind of dog you have). Because you must have these and you will use them frequently, be mindful of needing to replace them, which is wasteful and “un-green.” It may cost more initially, but buy quality tools that will last.
There are many other products you can use for grooming — shampoo, conditioner, hair accessories like bows, ointments, and solutions for cleaning eyes and ears, etc. So that you're not wasting these, think quality. Choose products that are first and foremost effective so you don't need to use as much of them, and then consider the source of the ingredients in the products you choose. Look for shampoos that are bio-degradable and free of dyes, alcohol, and petroleum products. While we enjoy scented products, dogs don't, and these are neither necessary nor healthy for your dog — especially “perfume” sprays.
If you prefer to have a professional groomer take care of your dog on a regular basis, ask to see what kinds of products they use. You can ask them to use what you'd like so long as you provide it.