Research and Development Tax Credit Expansion Act of 2021 This bill modifies the refundable research tax credit for new and small businesses to (1) increase the limit on refundability to $500,000, with an adjustment for inflation; (2) allow refundable amounts to cover all payroll taxes paid by such businesses; and (3) extend eligibility for the credit to small businesses with less than $10 million in gross receipts (currently, less than $5 million). The bill also increases the rate of the alternative simplified tax credit for such businesses.