You are not subject to the New York City unincorporated business tax (UBT) if your business is located outside of New York City and you conduct none of your business in New York City. It appears that you conduct all of your business outside of New York City, so the only question would be if the business itself is located in New York City. If you use your home address as the business address and/or you manage the business from your home, you would likely be subject to the UBT tax as that would mean the business was located there.
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