Veterans' Acute Crisis Care for Emergent Suicide Symptoms Act of 2020 or the Veterans' ACCESS Act of 2020 This bill requires the Department of Veterans Affairs (VA) to furnish emergent mental health care (i.e., emergency stabilization care) to eligible individuals in certain situations at VA or non-VA facilities. Specifically, the VA must furnish care if an individual seeks it (1) in an emergency situation, or (2) pursuant to a recommendation from the Veterans Crisis Line or licensed health care professional. Eligible individuals under the bill include veterans, persons enrolled in the VA health care system, and individuals who (1) were discharged or released from service under a condition other than honorable, (2) are not enrolled in the VA health care system, and (3) served for a period of more than 90 cumulative days. For individuals receiving care, the VA must ensure the Veterans Crisis Line notifies the Suicide Prevention Coordinator and the Office of Community Care. The VA must also determine the eligibility of the individuals for other VA programs and benefits and make referrals for follow-on care. The VA may not charge an eligible individual for emergent mental health care and shall provide reimbursement to individuals who receive care at a non-VA facility. The bill also expands eligibility for mental and behavioral health care for certain veterans who were discharged or released from service under a condition that is other than honorable and served in the Armed Forces for a period of more than 90 cumulative days.