You have English Ivy--also called California Ivy, Hedera helix. It is poisonous to your dog, if eaten. I do not know how much your puppy would have to eat. I would contact your Veterinary doctor and get his/her advice.
I am including the links for ASPCA on English Ivy and also, their list of all plants that can harm pets. You just have to click on the plant name on this list and it will tell you everything you need to know. A