WASHINGTON—The war in Ukraine has depleted American stocks of some types of ammunition and the Pentagon has been slow to replenish its arsenal, sparking concerns among U.S. officials that American military readiness could be jeopardized by the shortage. The U.S. has during the past six months supplied Ukraine with 16 U.S. rocket launchers, known as Himars, thousands of guns, drones, missiles and other equipment. Much of that, including ammunition, has come directly from U.S. inventory, depleting stockpiles intended for unexpected threats, defense officials say. One of the most lethal weapons the Pentagon has sent are howitzers that fire high-explosive 155mm...