Was this business a sole proprietorship or a corporation? In the latter case, typically, no one is personally liable for customer refunds.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship. You are strongly advised to contact an attorney to obtain legal advice before taking or refraining from taking any action with respect to the above.
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An answer to this question depends on the contracts involved (both between your husband and her mother) and with the customers. On a non-legal note, you also want to consider that this is his mother and take that into consideration when deciding what to do. Usually, when a business closes it must follow legal procedures to give notice to customers, or refunds and to resolve any debts. Most states may also require (if there was a Corp. or LLC) that the company dissolve with the state authority or it may keep getting charged for state corporate taxes. You should consult an accountant as to tax filing requirements.
Please note that this answer is not legal advice, and should not be relied upon. Each state has different state laws and procedures. A review of all documents and a consultation at a minimum is necessary to render effective legal advice. This answer does not create an attorney client relationship. Consult an attorney before you decide on a course of action.