Root and Stem Project Authorization Act of 2022 This bill authorizes the Forest Service and the Bureau of Land Management (BLM) to conduct certain forest restoration projects. Specifically, the Forest Service or BLM may enter into an agreement for a project on federal land that was developed through a collaborative process that meets local and rural community needs if the party with whom it enters into an agreement initially provides the Forest Service or BLM with a portion of the funding necessary to complete any analysis deemed necessary under federal law for consideration of the proposed project; the Forest Service or BLM uses the funding to pay a contractor included on a list of contractors that it maintains to conduct the analysis; upon completion of the analysis, the Forest Service or BLM solicits bids to carry out the project and enters into a contract or agreement under the Healthy Forests Restoration Act of 2003 to carry out the project; and using certain receipts from the project, the Forest Service or BLM repays the initially provided funding. For purposes of a civil action relating to such a project, any person that participated in the collaborative process to develop the proposal for the project shall be (1) entitled to intervene, as of right, in any subsequent civil action; and (2) considered to be a full participant in any settlement negotiation relating to the project. The authority to enter into an agreement and the requirement to maintain a list of contractors shall expire on January 1, 2033.