Manufacturing Jobs for Veterans Act of 2022 This bill requires the Department of Labor, as part of the Veteran's Workforce Investment Program, to implement the Veterans Manufacturing Employment Program to award competitive grants to three states for the establishment and administration of a State Manufacturing Employment Program. Such program must provide grants to manufacturing employers and joint-labor management organizations that provide eligible veterans with training, on-the-job training, apprenticeships, and training that leads to a recognized postsecondary credential. Eligible veterans are those who are employed by a manufacturing employer and enrolled or participating in a specified training, on-the-job training, apprenticeship, or certification class.