Iraqi interpreter Laith al-Haydar received multiple death threats for working with the American military at the height of the war in his country. In return for helping the U.S., he and tens of thousands of other Iraqis were promised U.S. immigration visas.

Nearly four years after he applied, the 41-year-old father of two is still waiting for a visa—and now faces a new setback: President Donald Trump signed an executive order suspending immigration from several countries with a Muslim majority, including Iraq, and a temporary...