Perry, you can surely pray for a sick pet as well as yourself.
I've heard of people doing it, but it's not common practice. The priesthood is meant for blessing God's children not nessisarilly their pets unless there's a specific purpose for it. It may be more appropriate for you to receive a blessing to help you though a tough time while dealing with your pets condition. Call the local bishop if you don't have any other contacts.
Thanks! Your answer is awaiting moderation.