Migrant Facility Transparency Act of 2021 This bill allows a municipality or county to block the establishment of a temporary influx facility within its borders and addresses other related issues. (Generally, these temporary facilities provide shelter capacity when there is a large increase in the number of migrants arriving into the United States.) Specifically, the Department of Health and Human Services (HHS) must, at least seven days before selecting a site for such a facility, provide certain information about the planned facility to specified federal, state, and local officials, including officials for the municipality or county where the facility will be located. If the municipality or county notifies HHS of its opposition to the facility within three days of receiving the information, HHS may not establish the facility at that location. HHS must also notify the relevant federal, state, and local officials of (1) certain changes related to such a facility's operations, and (2) any major security or safety incident at the facility. Within 14 days of this bill's enactment, HHS must report to Congress certain information about the safety of temporary influx facilities. If HHS does not meet this deadline, HHS must immediately pause all intake processing at such facilities until the report is submitted.