Defining "Animal"
Animals are are very broad subject to tackle with such a question. Are spiders animals? We will take the widest possible view to begin with, defining animals as any sort of being at a higher evolutionary level (evolution versus creation is another subject) than a rock. So the question is redefined as “Do biological entities besides humans have souls?”
The question obviously remains bigger than big. It resists efforts to limit it to a question and answer that can be easily understood by our brainpower. But why not give it a try? Many others before me have, so why not one more attempt?
God projected the entire creation “out” from Him, as vibrations. The “word” was God. Word meaning vibration. I believe that the current string theories of science are getting close to this understanding. They theorize that far smaller than atoms, the smallest "stuff" of creation is tiny vibrating “strings” of “something.”
Everything created, from the smallest vibrating strings to man, to the largest galaxies are all evolving spiritually back to the Source from which they came. Granted, this is taking a while. And included in this evolution are all the critters, even those not normally considered part of the “spiritual party.” Appearances can be deceiving. Through the practice of yoga, especially the practice of pranayama (such as Kriya yoga) appearances become more and more transparent, and the reality shines through.
God and Soul
All Creatures are Evolving Upwards
So, animals of all kinds are evolving “upwards” (a direction is a concept of the mind of course) back towards God. And the Christian bible says that God is keeping an eye on the animals. If so, would He just forget them at death? All is a part of God, and thus, when the unreality of this world is transcended, the false view that death brings about the end of our beloved pets when they pass away, ends, too. Do they have a human soul locked up inside? Do you think that a virus has a human soul locked up inside it? Or each and every atom evolving upwards? Does your best friend Joe, the bulldog? He may not be the backwards reincarnation of your former aunt Nelda, but he is someone you loved very much, and he has a soul energy which passes on. It is a vast universe, three part, physical, astral and causal (ideas). The answers to such questions are best found in following the way of understanding.