Pandemic Emergency Manufacturing Act of 2021 This bill establishes within the Department of Health and Human Services (HHS) an Emergency Office of Manufacturing for Public Health to manufacture and distribute medical products to address COVID-19 (i.e., coronavirus disease 2019) or medical products that are on shortage or vulnerable to shortage. The office must (1) obtain the rights to manufacture the applicable medicines, ingredients, diagnostic tests, medical devices, personal protective equipment, and supplies; (2) manufacture or contract to manufacture such items; and (3) construct, or enter into construction contracts for, facilities to manufacture certain medicines, such as vaccines. HHS may issue involuntary licenses allowing the office to make, use, sell, import, or export an invention related to an applicable drug, biological product, or device and to use clinical trial data and confidential information. Such a license must provide the rights holder with reasonable compensation. The office must (1) provide such COVID-19 products at no cost to federal, state, local, and other health programs and certain domestic health care providers and suppliers; (2) offer COVID-19 products at cost to other commercial and international entities; and (3) offer other medical products to entities at a fair price, based on cost and other considerations. The office shall prioritize the manufacture of certain COVID-19 products and meet manufacturing timelines specified in the bill.