Local Community Emergency Relief Act of 2020 This bill waives cost share requirements for all federal emergency declarations and major disaster declarations during 2020. Specifically, the bill requires the Federal Emergency Management Agency (FEMA) to provide a 100% cost share in such instances. In general, FEMA disaster assistance requires a 75% cost-share commitment from the federal government with 25% provided from nonfederal sources. The bill lists activities for which the federal cost share is waived, including miscellaneous costs, such as the costs of cleaning and sanitizing public areas and buildings where individuals who have contracted COVID-19 (i.e., coronavirus disease 2019) have been present to prevent the spread of the virus; other needs assistance for uninsured, disaster-related necessary expenses and serious needs, including specified child care assistance and funeral and burial assistance; mental health services; medical equipment; and hospital and morgue capacity expansion projects to meet actual or projected COVID-19 surge demands.