Accelerating Broadband Connectivity Act of 2021 This bill establishes the Accelerating Broadband Connectivity Fund from which the Federal Communications Commission (FCC) shall offer additional funding to certain terrestrial telecommunications carriers for specified broadband projects. Specifically, the FCC shall make one-time funding offers from the Accelerating Broadband Connectivity Fund to certain terrestrial telecommunications carriers that receive support from the Rural Digital Opportunity Fund. The Rural Digital Opportunity Fund is a mechanism through which the FCC finances high-speed broadband networks in unserved rural areas. Recipients of amounts from the Accelerating Broadband Connectivity Fund shall use such funds to (1) begin construction of a broadband network, (2) make broadband available from the network deployed using Rural Digital Opportunity Fund support, and (3) meet all build-out obligations pursuant to receiving amounts from the Rural Digital Opportunity Fund.