National Apprenticeship Act of 2021 This bill provides statutory authority for the registered apprenticeship program within the Department of Labor and for related grant programs. The bill provides statutory authority for the Office of Apprenticeship (OA) within Labor. The OA's responsibilities include (1) supporting the development of apprenticeship models; (2) recognizing qualified state apprenticeship agencies, and operating apprenticeship offices in states without a recognized agency; (3) providing technical assistance to state agencies; (4) periodically updating requirements for each occupation in the apprenticeship program and determining whether to approve new occupations for the program; (5) promoting greater diversity in the national apprenticeship system; and (6) awarding grants provided by this bill. The bill also establishes in statute the responsibilities of state apprenticeship agencies and offices, including (1) providing technical assistance to stakeholders, (2) resolving complaints, (3) establishing state performance goals, and (4) including in its written plan a description of how its apprenticeship programs align with the skills needs of the state's employers. The OA shall enter into an agreement with the Department of Education to promote the integration and alignment of apprenticeship programs with secondary, postsecondary, and adult education. The OA shall award grants, contracts, or cooperative agreements to eligible entities to (1) expand national apprenticeship system programs, including by expanding pre-apprenticeship and youth apprenticeship programs; (2) encourage employer participation; and (3) strengthen alignment between the apprenticeship system and education providers. The bill provides statutory authority for criteria for various programs, including (1) quality standards for apprenticeships, (2) requirements for apprenticeship agreements between a program sponsor and an apprentice, and (3) acceptable uses for grant funds awarded by this bill. The bill also provides statutory authority for the National Advisory Committee on Apprenticeships. The committee's duties shall include advising the OA on matters relating to this bill and providing recommendations on topics such as increasing the participation of populations not traditionally involved in the national apprenticeship system. Labor shall engage an independent entity to conduct research on ways to improve the management and effectiveness of national apprenticeship system programs.