The various military aid societies are stepping up to help families affected by the government shutdown. In addition to these specifics recently highlighted, they are always available to help with basic living expenses and emergency situations.
The Air Force Aid Society (AFAS) has announced that it is absolutely able to help military families who are not receiving their Women, Infants, and Children (WIC) benefits due to the government shutdown. Some AFAS offices are unavailable due to the shutdown. If you need assistance and your office is closed, you can contact any of the other aid societies or the American Red Cross for assistance.
The Navy-Marine Corps Relief Society (NMCRS) will continue to help with confidential loans and grants to cover basic living expenses and emergencies. NMCRS is not reporting any difficulties opening their offices.
Army Emergency Relief assistance has been expanded to cover new categories created due to the government shutdown. I have not heard any reports of people not being able to get AER help.
More information about the relief societies can be found in Military Relief Societies Can Help.