Federal agents have persuaded police officers to scan license plates to gather information about gun-show customers, government emails show, raising questions about how officials monitor constitutionally protected activity.

Emails reviewed by The Wall Street Journal show agents with the Immigration and Customs Enforcement agency crafted a plan in 2010 to use license-plate readers—devices that record the plate numbers of all passing cars—at gun shows in Southern California, including one in Del Mar, not far from the Mexican...