As I understand it... "Virginia income tax withheld" means you took out taxes to pay to the State of Virginia (Virginia State taxes). If taxes were taken out to be paid to the State of Virginia, then you must file the form 1099-MISC or 1099-NEC with Virginia. I don't believe this is a special form; it is one of the page sets found in the 1099-MISC / NEC that you send to the State. I would send the State copy of the form to the contractor as well (whether you took out Virginia income tax or not).
