Project Precursor Act This bill requires and authorizes actions to address fentanyl trafficking. The President must impose visa- and property-blocking sanctions on foreign persons (i.e., individuals and entities) knowingly involved in (1) the trafficking of fentanyl, fentanyl precursors, or related opioids by a transnational criminal organization; or (2) the activities of a transnational criminal organization relating to such trafficking. (Transnational criminal organizations are defined to include certain organizations, such as the Sinaloa Cartel.) The bill also authorizes the Department of State to carry out a program to build the capacity of Mexican law enforcement agencies to prevent the trafficking of fentanyl and its precursor chemicals into the United States; requires the State Department to prioritize Mexico in all efforts to combat the international trafficking of fentanyl and other synthetic drugs by providing technical assistance and equipment and carrying out exchange programs; requires the State Department to conduct a public relations campaign to delegitimize Mexican transnational criminal organizations involved in fentanyl trafficking and disrupt their ability to recruit new members; requires the State Department to use the voice, vote, and influence of the United States to include fentanyl substances on the lists of toxic chemicals identified in the Chemical Weapons Convention (CWC). (Chemicals on the lists are subject to the CWC's verification processes to ensure that such chemicals are not used for prohibited purposes.) Additionally, this bill extends the Fentanyl Sanctions Act to seven years after the date of enactment of this bill (the act is currently set to terminate on December 20, 2026).