Virtual Peer Support Act of 2020 This bill appropriates funding for grants to transition behavioral health peer support services that are provided at no cost to participants from in-person to virtual platforms or to otherwise expand these kinds of virtual services. Specifically, the Substance Abuse and Mental Health Services Administration must award competitive grants to certain consumer-controlled or consumer-run organizations and tribal communities. Among other eligibility requirements for these grants, entities must have offered certain behavioral health services at no cost to participants before the declaration of the COVID-19 (i.e., coronavirus disease 2019) public health emergency.