If $5,000 or less, you can qualify for a small claim. You need to buy an index number, file a summons and complaint. A guide to your local small claims court in Buffalo is here
If more than $5,000, you have to file in regular civil court, in which case I highly recommend you find a local lawyer to do it for you for part of the recovery although you can still do it yourself and find local resources to assist self represented litigation. An Erie county court guide can be found here
Finally, your theory is limited to breach of contract and fraudulent inducement of contract due to a material misrepresentation of a material fact - that the puppy was healthy. You want the contract rescinded or have her found in breach and subject to full disgorgement, plus expenses. If you do it yourself, keep it simple and just stay focused on what you can prove: that she sold you a defective puppy, agreed to pay you back, and now refuses. (If the puppy was contagious to humans or other animals, she also violated state law and other specialized statutes may apply to her "business" as an apparently licensed breeder. Legal aid is a good place to start for more detailed legal guidance.)
I agree with my colleague, but I urge you to seek the assistance of counsel as this may get sticky.
Also, if this person is located in another state, it may not be worth pursuing. Again, you should speak with a local attorney for further guidance.