Federal Land Freedom Act of 2021 This bill authorizes a state with an established oil and gas leasing program to take responsibility from the federal government for leasing and regulating the exploration and development of oil, gas, and other forms of energy on certain federal land in the state. In addition, the bill exempts state actions to lease, permit, or regulate oil and gas exploration and development from certain requirements under the Administrative Procedure Act, the National Historic Preservation Act, the Endangered Species Act of 1973, and the National Environmental Policy Act of 1969.