Student Apprenticeship Act of 2019 This bill requires the Department of Education to award grants to certain partnerships to enable the development and implementation of a postsecondary apprenticeship program. An eligible partnership must consist of (1) at least one institution of higher education, (2) at least one individual employer, and (3) a workforce intermediary (an entity that demonstrates expertise in fostering partnerships in order to support postsecondary apprenticeship programs). Such partnership may also include other entities such as a state agency responsible for administering career and technical education, a labor organization, or an economic development agency. A partnership that receives a grant must use the funds to develop and implement a postsecondary apprenticeship program in a high-skill, high-wage, and in-demand industry sector or occupation. Additionally, an apprenticeship shall qualify as a work-study program through which a student apprentice is eligible for federal financial assistance.