Protect American Election Administration Act of 2022 This bill generally prohibits a state from soliciting, receiving, or expending any payment or donation of funds, property, or personal services from a private entity for the purpose of administering a federal election. For example, the bill prohibits the use of these payments or donations for programs related to voter education, outreach, and registration. The prohibition shall not apply to a state's acceptance and use of a private entity's donation of space to be used for a polling place or an early voting site.