Apprenticeship Access for All Act of 2021 This bill requires the Department of Labor to promote diversity and ensure equal opportunity to participate in programs for apprentices, youth apprentices, and pre-apprentices by taking steps to promote diversity in apprenticeable occupations under the national apprenticeship system, especially in high-skill, high-wage, or in-demand industry sectors and occupations in areas with high percentages of low-income individuals; ensuring programs under the national apprenticeship system adopt and implement policies for equal opportunity in such programs, do not engage in prohibited intimidation or retaliation, and are subject to enforcement action; and supporting the recruitment, employment, and retention of nontraditional apprenticeship populations in high-skill, high-wage, and in-demand industry sectors and occupations, including women, people of color, individuals with disabilities, individuals impacted by the criminal and juvenile justice system, and individuals with barriers to employment.