Sorry, but no. If animals had souls they would also need to have free will and cognitive ability and they would be subject to the laws of God. They are instinctual and are subject to the laws of nature.
I don't think there is any mention of it in the bible .. So I believe it's just simply unknown. I just know the they possess as much 'Free Will" and cognitive behaviour as people do, not to mention are capable of an emotional level far greater than we, as humans.
The only time "Free Will" is lacking is when we humans, take it away. Animals in the wild have very instinctual behaviours .. they need to in order to survive .. but they possess love and devotion in the purest form, hate, the ability to mourn, disappointment and even can be happy or sad.
I don't know if they have souls .. but they have spirit. They possess pureness and are not contaminated with all the crap we as humans are. The only reason an animal "goes bad" (so to speak) is when they are conditioned to do it from abuse.
every living creature definitely has a soul, they are all God's creations, if God couldn't possibly hang with them in heaven after their deaths, why create them? it would just take too much time and effort just for nothing