Yes, animals have souls. Humans are made up of BODY, SOUL and SPIRIT. Animals of BODY and SOUL. Plants of BODY.
The physical body dies. We see that in plants, animals and humans.
The soul is the personality, will, emotions that make up who you are. If you lose a leg, you are still you. Animals have personality, will, emotions, intellect, ability to reason, but they were not made in the image of God.
Our spirit is made up of (for lack of a better way to put it) the same "material" that God is made up of. It's this spirit that will carry our soul after the physical death. The spirit will either carry the soul to God, or away from Him. Animals don't have a spirit but they do have souls.