Supporting Emergency Responders Volunteer Efforts Act of 2013 or the SERVE Act of 2013 - Amends the Internal Revenue Code to allow an individual taxpayer who is a bona fide volunteer of a qualified volunteer emergency response organization a $1,000 refundable tax credit.  Requires such individual to: (1) have served as a bona fide volunteer performing fire fighting and prevention services, emergency medical services, and ambulance services for more than six months in a taxable year; (2) have provided more than 40 hours of such services actively engaged in the prevention, control, or extinguishment of fires or response to emergency situations where life, property, or the environment is at risk; or (3) have been stationed on the premises of an emergency response organization in anticipation of being actively engaged in providing such services.