Unemployment Insurance Systems Modernization Act of 2021 This bill expands the list of requirements a state unemployment compensation system must meet for the state to receive federal funds for administration of the system. Specifically the bill requires the unemployment law of each state to include various provisions to promote adaptability, including provisions for (1) handling surges in claims, (2) processing disaster unemployment assistance claims, (3) processing claims under temporary federal benefits programs, and (4) automating claims for short-time compensation. The bill also requires states to utilize databases and other systems in an effort to prevent fraudulent and improper unemployment compensation payments. Specifically, states must use (1) a system selected by the Department of Labor for cross-matching unemployment claimants against available databases, (2) an automated system developed and approved by Labor for exchanging unemployment compensation information, and (3) the existing National Directory of New Hires for identifying individuals who have become employed.