According to an Army Times article, commissaries will be closed if a government shutdown occurs. (There is a possibility that some overseas stores will remain open.)
Additionally, a news release issued by the Army and Air Force Exchange Service (AAFES) states that the Exchange (PX/BX) will remain open if the government shuts down.
"While the federal government shuts down, business at the Exchange remains largely unchanged. However, some transactions may be delayed, such as the purchase of firearms which require background checks or other federal government actions," said Col. Virgil Williams, Chief of Staff for the Exchange.