Yes, you can get poison ivy from your dog, cat or any other outdoor pet. Since animals' fur protects their skin from the urushiol oil, they typically don't develop a rash. It can be difficult to tell if they have been in contact with poison ivy or oak plants, or carrying the rash-causing oil. However, the oil will remain on their fur and may contaminate you when you touch them.
A safe way to remove the rash-causing oil is to wash your dog or cat with Tecnu Original Outdoor Skin Cleanser. It is safe to use on your pets to remove the oil from their fur, but should always be followed with a pet bath to make sure all of the Tecnu is removed.