Apprenticeship and Jobs Training Act of 2019 This bill allows employers a business-related tax credit for up to $5,000 for the training of a qualified individual in a qualified apprenticeship program or multi-employer apprenticeship program. A "qualified individual" is an individual who (1) is an apprentice participating in a qualified apprenticeship program or multi-employer apprenticeship program, (2) has been employed in either program for a period of at least seven months that ends within the taxable year, and (3) is not a highly compensated employee or a seasonal worker. A "qualified apprenticeship program" is a program that (1) provides qualified individuals with on-the-job training and instruction for a qualified occupation (i.e., a skilled trade occupation in a high-demand mechanical, technical, health care, or technology field); (2) is registered with the Office of Apprenticeship of the Department of Labor or a state apprenticeship agency recognized by the office; and (3) maintains records relating to the qualified individual. A "qualified multi-employer apprenticeship program" is a program in which multiple employers are required to contribute and that is maintained pursuant to one or more collective bargaining agreements. The bill also allows certain distributions, without penalty, from retirement and pension plans to an employee who is serving as a mentor. A "mentor" is a working individual who (1) has attained age 55; (2) works reduced hours and engages in mentoring activities for at least 20% of such hours; and (3) is responsible for the training and education of employees or students in an area of expertise for which such individual has a professional credential, certificate, or degree.