Candy Hill in Winchester, VA has storage area. May be a little to far away for you. Check out the campgrounds in the area where you are staying. Most campgrounds also have a storage or dry camping area.
There are two or three campgrounds I know of in the Luray area. I would think one or more would have storage facilities. One is Jellystone Park. I forgot the name of the other but could look that up if you need the info. Or do a google search on luray campground. Email (use the envelope icon above) me if you need me to look it up for you.