Reopening America by Supporting Workers and Businesses Act of 2021 This bill permits states to provide a one-time lump sum payment, through July 1, 2021, to individuals who (1) were eligible for COVID-19 pandemic unemployment compensation during any week after enactment of this bill, and (2) are no longer eligible as a result of becoming thereafter reemployed. Payments of $1,200 shall be paid to individuals who work at least 30 hours and $600 to individuals who work at least 20 hours but less than 30. The payments are available for claimants (1) who are not employed by a government entity, (2) who return to work for at least four consecutive weeks, and (3) whose annual salary does not exceed $75,000. Before receipt of a payment, the individuals' employers must verify their hours and earnings. Additionally, the bill (1) accelerates a scheduled increase in funding for reemployment services and eligibility assessments (RESEA), (2) allows states to use RESEA funds to serve all workers receiving unemployment benefits, and not just those most likely to exhaust their benefits; and (3) allows states to provide reemployment services to individuals receiving pandemic unemployment assistance and pandemic emergency unemployment compensation. Finally, the bill reinstates the requirement that unemployment claimants certify they are able, available, and actively seeking work.