Foreign Political Influence Elimination Act of 2021 This bill expands the prohibition on campaign spending by foreign nationals. It also revises foreign-agent disclosure requirements. Specifically, the bill expands the ban on campaign spending by foreign nationals to include corporations that are subject to specified levels of ownership or control by foreign nationals or governments. The bill prohibits certain types of tax-exempt, foreign-affiliated organizations from taking certain campaign-related actions, such as contributing to a super political action committee. Further, the bill prohibits foreign nationals from (1) participating in the decision-making process regarding an election expenditure, or (2) making contributions in connection with state or local ballot initiatives or referenda. The bill prohibits political committees from accepting direct or bundled contributions from registered foreign agents. Next, the bill makes various changes to the Foreign Agents Registration Act of 1938 (FARA), such as by (1) repealing an exemption from FARA requirements for individuals registered as lobbyists, and (2) expanding filing and labeling requirements related to the transmission of informational materials in the interests of a foreign principal. In addition, the bill provides the Department of Justice (DOJ) with a mechanism to demand evidence for investigating compliance with FARA. Further, the bill provides for various civil penalties for failing to meet agent registration requirements. The foreign principal of a penalized agent may not pay the imposed fines. DOJ must establish a unit for the investigation and enforcement of FARA. The Government Accountability Office must analyze the effectiveness of FARA enforcement.