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.