Border Business COVID-19 Rescue Act This bill establishes and provides funding for a program whereby the Small Business Administration (SBA) must make loans to border businesses directly impacted by COVID-19 (i.e., coronavirus disease 2019). A border business is any entity, otherwise eligible for an SBA disaster loan, that (1) has its principal office in the United States, (2) has annual revenue of up to $500,000, and (3) is located within 25 miles of the U.S. border. The SBA shall make loans of up to $500,000 to border businesses, and these loans must have a zero percent interest rate. Loan recipients must use the funds to mitigate the effects of COVID-19 on their business, including by (1) providing paid sick leave to employees unable to work because of the pandemic, (2) making rent or mortgage payments, (3) purchasing personal protective equipment, or (4) paying for logistical expenses associated with border closures due to the pandemic. Recipients may also request an advance on the loan, which shall not be less than $10,000. Further, certain recipients may be forgiven up to 100% of the loan, less any amount received under other specified COVID-19 economic relief programs.