"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.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
