I really think this verse should be read in the context of Genesis 9:1-6. It has no relevance at all to the original question.
This verse is God telling Noah that animals are going to fear man now. That man can eat both plants and animals now, but man isn't allow to eat them while they are alive. He must kill the animal first. Otherwise, this is cruelty. And the punishment for eating an animal alive is for you to be killed at the hands of man or another animal. And if someone shed's man's blood, he'll be killed by man. As far as horse-racers, hare-hunters, etc. goes...that's stretching it, and I read all of Chapter 9. After this command, God makes the covenant with the rainbow.
That verse doesn't have anything to do with animal's sin.
So to answer the original question, think of what sin is. Sin is disobedience to God. Can animals disobey God? Do they have His commands written on our heart like we do? (Yes, every man has God's commands written on their heart. Untouched tribes in Papua New Guinea have a form of the 10 Commandments in their tribes.) I don't know what the first four commandments are to an animal, though animals have killed their own kind over territorial battles and whatnot. They sleep around to procreate. Some animals kill their parents, and some parents kill their young. Not enough info on animal language to know if they lie. We know they steal food sometimes (hence the puppy dog face)...
But you know, thinking about a dog that feels guilty, does it feel guilty really? Or perhaps the dog has been trained not to do what it did, did it anyway out of instinct, and now is worried about the punishment. There's no guilt of "I did something wrong", but instead the conditioned response of fear at being yelled at, bopped on the nose, or maybe getting a slap on the butt. Just as animals can be trained to do tricks or stay out of the other pets' food dishes, they can be trained to know when they're about to get in trouble.
Without humans training them how to obey, and thus how to disobey, they would be wild and do what they pleased. Yet humans, without being "trained" the laws of God, have His law written on their heart, and you will find some obeying, yet many disobeying, even in tribes that have never come in contact with anyone other than themselves and perhaps one or two neighboring tribes.
Animals can't sin. If they could, it would bring into account, "Do they go to heaven or hell?" And that would lead to, "Okay, well plants are alive to. Do they go to heaven or hell?" I think humans are a bit more special than to get lotted up with the animals and plants.