Federal Extreme Risk Protection Order Act of 2023 This bill authorizes and establishes procedures for federal courts to issue federal extreme risk protection orders. Additionally, the bill establishes grants to support the implementation of extreme risk protection order laws at the state and local levels, extends federal firearms restrictions to individuals who are subject to extreme risk protection orders, and expands related data collection. Extreme risk protection order laws, or red flag laws, generally allow certain individuals (e.g., law enforcement officers or family members) to petition a court for a temporary order that prohibits an at-risk individual from purchasing and possessing firearms. Among its provisions, the bill authorizes a family or household member, or a law enforcement officer, to petition for a federal extreme risk protection order with respect to an individual who poses a risk to themselves or others; directs the Department of Justice to establish a grant program to help states, local governments, Indian tribes, and other entities implement extreme risk protection order laws; extends federal restrictions on the receipt, possession, shipment, and transportation of firearms and ammunition to individuals who are subject to extreme risk protection orders; and requires the Federal Bureau of Investigation to compile records from federal, tribal, and state courts and other agencies that identify individuals who are subject to extreme risk protection orders.