Mobile Health Clinics Act of 2019 This bill permits the Department of Health and Human Services (HHS) to establish a grant program to increase mobile medical health care services for medically underserved communities and medically underserved populations. (These are geographic areas and populations designated as lacking access to primary care services.) HHS may provide grants to partnerships of (1) one or more hospitals; or (2) one or more local health care facilities, including clinics, federally qualified health centers, or pharmacies. Among other purposes, grantees may use funds to purchase or maintain mobile health care services vehicles; hire and train staff; and increase access to required primary health services, substance use disorder services, and mental health counseling.