Flood Mapping Modernization and Homeowner Empowerment Pilot Program Act of 2021 This bill directs the Federal Emergency Management Agency (FEMA) to carry out a pilot program to make grants to local governments to enhance the mapping of urban flooding and associated property damage and the availability of such mapping data to homeowners, businesses, and localities to enable them to minimize flooding risk. The grants may only be made to local governments in urbanized areas with populations greater than 50,000 or to stormwater management authorities in those localities. FEMA shall give priority to grant applicants in areas that (1) are highly vulnerable to a sea level rise, (2) include a military installation or other national security facility, or (3) have a population highly vulnerable to urban flooding and have an uneven capacity for flood mitigation and response efforts due to socioeconomic factors.