"Can I use form 763S to request a refund of the taxes withheld without actually filing a Virginia tax return?"
Yes. If you were a full-year non-resident of VA with no VA-source income to report, then you are not required to file a VA tax return. Note that VA Form 763-S cannot be e-filed; it must be printed and mailed.
And get that fixed.....You will owe NC taxes on it.
See if they can withhold NC taxes instead for the future.
IF they can't withhold for NC, then you will likely need to arrange to pay NC quarterly estimated taxes instead.
I am a nonresident of Virginia, the VA withholds taxes from my real estate investment income. Which form should I use to claim excess tax withholding, 763 or 763s?