Library Stabilization Fund Act of 2020 This bill establishes and provides funds for a Library Stabilization Fund to assist libraries with responding to and recovering from the coronavirus (i.e., the virus that causes COVID-19) pandemic. Specifically, the bill directs the Institute of Museum and Library Services (IMLS) to allocate funding to states, Indian tribes, and organizations that primarily serve and represent Native Hawaiians. From these allocated funds, the IMLS must also make emergency library relief grants to state library administrative agencies with approved five-year plans, Indian tribes, and organizations that primarily serve and represent Native Hawaiians. Additionally, the IMLS must allocate funding to carry out activities that strengthen the ability of libraries to provide services to communities affected by the coronavirus pandemic. The IMLS may enter into arrangements (e.g., grants and contracts) and other forms of assistance with libraries, library consortia and associations, and institutions of higher education to carry out these activities. Funds allocated by the bill may be used to provide support to libraries, such as by supporting general operations; paying staff; purchasing cleaning supplies; training library staff on the safe handling of library materials; providing access to computers and other technology; purchasing digital resources; expanding resources related to early education and literacy, distance learning, and workforce development training; and ensuring access to community services and activities.