Veterans-Specific Education for Tomorrow's Health Professionals Act or the VET HP Act This bill requires the Department of Veterans Affairs (VA) to implement a pilot program, located at a minimum of five VA medical centers, to provide a clinical observation experience to students who are enrolled at an institution of higher education that has referred them to the program. The VA shall give priority to students who (1) resided in a health professional shortage area, (2) are first-generation college students, (3) have been referred by minority-serving institutions, (4) are veterans, or (5) indicate an intention to specialize in an occupation with a health professional shortage.