The conditions under which POD beneficiary designations are effective are purely a function of state law. Many states describe their provisions in a "multiple party" account statute. In one state, the provisions are actually contained in the state's probate code. A generic answer will be of no value to you.
Your state bankers association probably offers seminars on opening new accounts or deposit account administration that contain this information.
Note that even if such a designation is possible, it simply might not be a very smart thing to do if the sole proprietorship is a going business with employees and accounts receivable to pay. Doing what the customer wants could yield disastrous circumstances for the continued operation of the business on the owner's death.