As a general rule pepper spray may be carried. Below you will find two Virginia Code Sections regarding pepper spray you should be aware of when carrying pepper spray.
You may also want to consult with your local law enforcement to determine if there are any local ordinances that may be applicable. Also be aware that federal property and federal facilities are governed by federal law, and you should contact the various agencies to determine what if any restrictions apply.
§ 18.2-312. Illegal use of tear gas, phosgene and other gases.
§ 18.2-52. Malicious bodily injury by means of any caustic substance or agent or use of any explosive or fire.