Promoting Rigorous and Innovative Cost Efficiencies for Federal Procurement and Acquisitions Act of 2021 or the PRICE Act of 2021 This bill directs the Management Directorate of the Department of Homeland Security (DHS) to publish an annual report on a DHS website on projects that have used innovative procurement techniques within DHS to accomplish specified goals. Such goals are improving or encouraging better competition, reducing time to award, achieving cost savings, achieving better mission outcomes, or meeting the goals for contracts awarded to small business concerns. The Management Directorate shall (1) develop and disseminate guidance and offer training for specified personnel concerning when and how to use such techniques, and (2) share best practices across DHS and make available to other federal agencies information to improve procurement methods and training. The Office for Federal Procurement Policy shall convene a Chief Acquisition Officers Council to examine best practices for acquisition innovation in contracting in the federal government.