A non-VA CCW permit won't help. However, this being a shall-issue state, the process is simple enough. I am not sure about other locations, but in Fairfax County you have to apply in person at the Circuit Court. No appointment necessary. After approval, the permit will be mailed to you.
https://www.fairfaxcounty.gov/circuit/civil-case-information/concealed-handgun-permit
I used NRA Basic Pistol Course certificate as my proof of handgun competency.
You can get the application on line and do the fingerprinting at the VASP building over in Fairfax for $10 cash or check. Mail in all forms with copy of "competency" to VASP in Richmond and then wait.