This New York Times article says:
"[T]hree states besides Connecticut — Alabama, Virginia and Tennessee — as well as other jurisdictions, prohibit weddings performed by ministers who do not have active ministries.
Perhaps the reference is to Va. Code 20-23, which says: "When a minister of any religious denomination shall produce before the circuit court of any county or city in this Commonwealth, or before the judge of such court or before the clerk of such court at any time, proof of his ordination and of his being in regular communion with the religious society of which he is a reputed member, or proof that he holds a local minister's license and is serving as a regularly appointed pastor in his denomination, such court, or the judge thereof, or the clerk of such court at any time, may make an order authorizing such minister to celebrate the rites of matrimony in this Commonwealth. Any order made under this section may be rescinded at any time by the court or by the judge thereof."
But, there is also Va. Code 20-25, under which "Any circuit court judge may issue an order authorizing one or more persons, resident in the circuit in which the judge sits, to celebrate the rites of marriage in the Commonwealth. Any person so authorized shall, before acting, enter into bond in the penalty of $500, with or without surety, as the court may direct."
I once heard that a certain bearded Assistant United States Attorney obtained such an order and performed a marriage ceremony.
Section 20-25 also says: "Any judge or justice of a court of record, any judge of a district court or any retired judge or justice of the Commonwealth or any active, senior or retired federal judge or justice who is a resident of the Commonwealth may celebrate the rites of marriage anywhere in the Commonwealth without the necessity of bond or order of authorization."
So, you could get a judge, if he or she was willing, to perform your rites of marriage anywhere in the Commonwealth.
UPDATE: A reader tells me his wedding was performed by a justice of the Virginia Supreme Court, and he is also aware of a wedding presided over by a Virginia sheriff.