For more than a year, Qatar has been locked in a diplomatic feud with Saudi Arabia, the United Arab Emirates, and other Middle Eastern nations, which blockaded their Persian Gulf neighbor over its close ties to Iran and alleged support for terrorism. This fight has spilled over into Washington, where the small but gas-rich country has mounted an aggressive influence campaign to counter its regional rivals—one that has involved not just an army of high-paid lobbyists but also allegations that Qatar hacked and leaked the emails of a top Republican fundraiser and Emirati ally viewed as an impediment to Qatar’s charm offensive.