Can my lawyer send me a copy of my signed separation document?

The question is not whether it is illegal, but whether it is unethical. I am a Florida lawyer, not a Virginia lawyer, but in Florida, so long as the client has paid the legal fees, the lawyer has no right to withhold the client's file, including agreements. You do have a couple of alternatives, first, you can contact the VA state bar association and file a complaint or you can go to the courthouse and obtain a copy of whatever it is you need. Good luck!