First of all this isn’t for me as I don’t live in either state

Can someone in TN sell a rifle to a VA without going to an FFL?  I said no but wanting to ask for clarification

I'm not sure how the law is for stuff like handing down family guns ( i know u didn't ask), but if complete strangers just meet at a Virginia gun show and just have the VSP background check and be done with legal requirements......no FFL

Don’t have answer to OP’s question but does it depend on type of firearm?

I bought a bolt action rifle at a gun shop in PA several years ago. I was a VA resident driving by a PA shop and decided to stop to look. Found a rifle I wanted and the guy showing me the rifle said he was not sure if he could sell it to me or not. The guy and and another guy, I guess he was the manager, talked about it and it was decided because it was a bolt action they could sell it to me. They said they could not sell anything considered to be an assault rifle to me as I was a VA resident. No clue, was just glad to be walking out the door with a rifle I had been looking for.

Back then they could've sold you an AR or any long arm and been legal with national laws.  Since 01 Jul 2020 Virginia has a background requirement for any firearm, thanks Coonman.  I eluded to family transfer which I am not certain on; just fishing on this thread to see if there are exemptions.

No.  Interstate firearms transactions have to go through an FFL.
I'm sure there are some obscure exceptions but generally, No.
VA residents have the UBC law so they have to receive all firearms through an FFL anyway.

TYCOM