Final answer:
The award originator should request the Federal Bureau of Investigation (FBI) to conduct record checks.
Step-by-step explanation:
The agency that the award originator should request to conduct record checks is the Federal Bureau of Investigation (FBI). The FBI is responsible for conducting background checks and investigating criminal activities that could potentially cause embarrassment to the United States if a foreign military personnel award nominee has engaged in such activities. By conducting record checks, the FBI ensures that deserving individuals are awarded without any detrimental consequences.