I love this question!
Yes, animals have souls. Buddhists believe that the intangible immeasurable portion of a person that makes up what is best described as a soul does not simply vanish when they are gone but it redistributed and moved into a new vessel. This concept is widely known as reincarnation.
Many Buddhist practitioners interpret this belief to mean that our souls are transmitted directly into the bodies of other living beings such as newborn children or animals after death. With this in mind it is easy to assert that many, if not all practicing Buddhist, would believe that animals have souls. There are always people who have differing beliefs within any organized religion. Some people believe that the transference of souls after death is not done in total but rather in a piecemeal type fashion. Some believe that souls are transmitted in a more indirect way similar to how water in a shallow pond is converted to vapor first, then rainclouds, then rain, then rivers, then finally the ocean. The idea of reincarnation may seem outlandish at first, but I like to think of it from a perspective of logic that my science background has given me.
It goes a little something like this: let us say that there is something within us that we have not yet found a way to measure that gives us our personality, our will, our capacity for love, and our individuality. Let us call this thing “soul.” If a soul exists (that is a big if for some of the skeptics in the readership) this soul is composed of matter or energy or some combination of the two. Then it is governed by the same physical laws that govern the rest of the known universe, namely the laws of conservation that state that matter and energy cannot be created nor destroyed, they can only change form. Let us argue that the soul is made from some third element that is neither energy nor matter; in that situation the hypothesis would be that this third element may also conform to the laws of conservation that have been established for matter and energy. But I digress. The question is about souls in animals. It is my personal belief that we are animals, we possess something within us that makes us more than the sum of our parts and that other animals have this quality as well. For lack of a better term this something is called a soul. I believe both humans and other forms of animals possess a soul, and it is a link between us and animals that should be celebrated.