Fair and Open Competition Act This bill prohibits a federal executive agency that awards any construction contract after the enactment of this bill from requiring or prohibiting a contract bidder from entering into agreements with labor organizations (i.e., Project Labor Agreements [PLAs]) or otherwise discriminating against a bidder or contractor who signs, or refuses to sign, a PLA. Agencies that award grants, provide financial assistance, or enter into cooperative agreements for construction projects after the enactment of this bill must ensure that the bid specifications, project agreements, or other controlling documents for such projects do not contain any requirements or prohibitions relating to PLAs. An agency may exempt a particular project or grant from the prohibition of this bill if it determines that special circumstances exist requiring an exemption to avert an imminent threat to public health or safety or to serve the national security.