Terror Intelligence Improvement Act of 2019 This bill requires each federal department or agency to provide to the Federal Bureau of Investigation (FBI) information about a person who is or has been under a federal terrorism investigation. The FBI and its Joint Terrorism Task Forces must immediately be notified of a request to transfer a firearm or explosive to a person who is, or was within the previous 10 years, under a federal terrorism investigation. The Inspector General of the Intelligence Community must audit the terrorism-related screening and watch list procedures of the federal government. The bill directs the Department of Justice to establish a process to delay and prevent the transfer of a firearm or explosive to a person who is, or was within the previous 10 years, under a federal terrorism investigation.