In some states, you can gain access to a safe deposit box by filing a petition to the court. You don't necessarily have to probate the case. Call the probate court in your county and ask if this is possible. Good luck.
That is a difficult problem. She may have an estate hidden in the safe deposit box.
One way around this problem could be Virginia's "Will search" exception. A Will search is allowed where there is a belief that the decedent's Last Will and Testament may be stored in the safe deposit box. It isn't good for gaining full access to the box, but at least you would know whether or not anything was stored in the box.
Since your grandmother died so long ago, a Will search may not be a viable option. This being the case, your simplest option is probably to manage it through probate.
Since your grandmother apparently has no assets, then you may qualify as a small estate under Virginia law. You should let the Clerk of the Circuit Court where you qualify know that there are no assets known of, and that you simply are getting the authority to access the safe deposit box.
If your grandmother had a Will, then it should be put to record and person named as executor (or personal representative) should go to the Circuit Court Clerk's office and qualify as the executor for this purpose. Alternatively, if the person(s) named does not want to qualify (or us unable to do so) then someone not named may qualify as the administrator c.t.a. If there was no Will, then any interested party may qualify as the administrator.
Although probate may seem like an arduous task, this "small estate" probate should be a very simple process, although it may be a bit time consuming.
I am licensed in Virginia and I would be happy to help you find a local attorney in Richmond if you need such assistance. Also, the Virginia State Bar website (www.vsb.org) has an attorney referral service if you need help. I have answered your question with general information, and this is not intended to be legal advice. No attorney-client relationship is established or should be expected as a result of my giving this general information.