The U.S. Farm Belt helped deliver Donald Trump to the White House, drawn to his promises to revive rural America and deregulate industry. Now, the president’s global trade offensive is threatening the livelihoods of many farmers.

Mounting trade disputes, spurred by U.S. threats to withdraw from the North American Free Trade Agreement and tariffs on billions of dollars’ worth of goods from key trading partners, have cut U.S. agricultural exports and sent commodity prices tumbling. Many farmers, who depend on shipments overseas...