When the young man came by her office last spring, Cara O’Callaghan thought he was one of the students she sees regularly in the recreation department at the University of California, Santa Barbara, where she’s been a finance manager for nearly 10 years. The UCSB graduate and mother of a teenager helps students manage their club teams’ budgets. She prizes the one-on-one time with them.

Her delight soon turned to frustration: The man wasn’t a student but an organizer for Teamsters Local 2010, a union representing thousands...