Yes, the written agreement can be introduced in court without notorization. Very few written contracts are notorized. The question is- do you want to spend the money going to court to get $60.00? I don't know what small claims costs in Kentucky, but in Maryland where I practice it would cost well more than $60.00 including filing fees, service of process fees, and fees for execution of judgment.
I agree with Attorney Mauro. There is no requirement that a contract be notarized, in order to be enforceable. If your family member denies the signature in court, it will depend on who the judge believes. I think that based on the facts you have stated, you have a very good chance of winning a small claims action. The practical problem will be collecting that judgment. The family member may not be collectible. If she cannot pay you $60 for your puppy, how are you going to collect a judgment? I am also not sure it is worth the time and hassle to take her to court for $60. Even if you can collect court costs, as well, you are still talking about at least a couple hours worth of time and the stress of dealing with the court. I would personally rather have the dog.
I am licensed to practice law in the State of Michigan and have offices in Wayne and Ingham Counties. My practice is focused in the areas of estate planning and probate administration. I am ethically required to state that the above answer does not create an attorney/client relationship. These responses should be considered general legal education and are intended to provide general information about the question asked. Frequently, the question does not include important facts that, if known, could significantly change the answer. Information provided on this site should not be used as a substitute for competent legal advice from a licensed attorney that practices in the subject area in your state. The law changes frequently and varies from state to state.