Whenever a question involves property rights, including questions like this, it's always a good idea to check out state law. That said, if your state recognizes Pay on Death designations, it's probably possible for a sole proprietor to put a POD designation on his or her sole proprietorship account.
It's important to note, however, that the POD designation only addresses the deposit account. It does nothing about transferring ownership of the business itself, or any of the equipment, supplies, etc., used in the business. The farmer obviously has to make other arrangements for transferring that type of property, or there won't be any business for his son to run.