Restoring Essential Energy and Security Holdings Onshore for Rare Earths Act of 2022 or the REEShore Act of 2022 This bill requires the inclusion of rare earth elements and certain other critical minerals in the National Defense Stockpile and sets out other requirements related to those elements and minerals. Specifically, the bill requires defense contractors to disclose the provenance of permanent magnets that contain those elements or minerals and expands prohibitions that restrict the Department of Defense from procuring goods or services, including those elements and minerals, from companies affiliated with China. The bill also expands the President's authorities under the Defense Production Act of 1950 to support domestic sources for those elements and minerals. (The Defense Production Act of 1950 confers on the President a broad set of authorities to influence domestic industry in order to provide essential materials and goods needed for the national defense.)