If VA taxes were withheld from your pay, you're correct that you would have to file a VA tax return in order to get those refunded.
If you have to file a non-resident NY return, be sure to complete that first, before you do your home state VA return. You can find NY's filing requirements for non-residents here:
https://www.tax.ny.gov/pit/file/nonresidents.htm
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
