My military pay was taxed in the wrong state where I wasn't a resident of this state for the entirety of 2022. How would one ask for this money back

In most cases, you have to file a nonresident tax return and report zero state wages to get a refund of withholding that was incorrect.  However, Virginia has a separate form to request the refund.  You can complete the 2022 Form 763S, Virginia Special Nonresident Claim For Individual.  Be sure DFAS has the correct Home of Record so the withholding is correct and maintain your state residency status.