MUHAYMIDA, Syria—In Islamic State’s former eastern Syrian stronghold, the Islamic Republic of Iran is parlaying its military and economic might into a permanent foothold.

On the heels of an Iranian military intervention that has helped bring President Bashar al-Assad to the edge of victory in Syria’s eight-year-long war, Tehran is moving to cement its long-term influence in Syria by cultivate goodwill and wining converts to the Shiite Muslim sect. To Syrians battered by war, Iran is offering cash, food, Iranian ID cards,…