Omnibus Travel and Tourism Act of 2021 This bill addresses measures to support the U.S. travel and tourism industry with a specific focus recovery from the COVID-19 pandemic. The bill establishes the office of Assistant Secretary of Commerce for Travel and Tourism. Among other responsibilities, the Assistant Secretary must develop and implement a strategy to assist the U.S. travel and tourism industry to recover from the COVID-19 pandemic. The bill also provides statutory authority for the United States Travel and Tourism Advisory Board and directs the board to assist with the recovery strategy. Additionally, the Department of Commerce must study the effects of the COVID-19 pandemic on employment, revenue, and other aspects of the travel and tourism industry and provide policy recommendations to promote the industry. Commerce must also publish trends in domestic travel and tourism. Further, the Department of Health and Human Services must establish a joint federal task force to address health, safety, security, and other logistical issues affecting air travel during and after the COVID-19 pandemic. The bill establishes a committee comprised of representatives of airports and other aviation stakeholders to advise the task force. In addition, the Transportation Security Administration (TSA) must study the feasibility of using dogs to detect the presence of the virus that causes COVID-19 as part of airport security screenings. The TSA may also establish a pilot program at up to six foreign airports to allow passengers departing from those airports to bypass domestic security rescreening at a connecting airport in the United States.