Victor Gutierrez was working his way through community college by balancing two dead-end jobs when he heard about an apprenticeship that would enable him to finish his coursework while earning a paycheck. "It fit perfectly," he said. But Mr. Gutierrez, 18 years old, isn't studying welding or auto mechanics. He is majoring in business management—and the apprenticeship is in a landmark skyscraper that houses Aon PLC, one of the largest professional-services companies in the world. In addition to free tuition and training, the Chicago resident earns $38,500 a year. He is among a small but growing number of young people apprenticing in fields that require a suit and tie instead of coveralls and gloves.