Never heard of Robyn Hurder? Don’t be alarmed.

The 35-year-old actress from Maine has appeared in the lead female role of Roxie Hart in the long-running Broadway production of “Chicago” a mere six times since joining the show this past summer. As an understudy, her presence is typically announced unceremoniously with a slip of paper inserted into the program.

Yet when Sloane Kelly, a New York high-school student, bought a ticket to “Chicago,” she purposely chose a night when Ms. Hurder would be performing. “She is just so passionate about the role,” Ms. Kelly explained.

Broadway’s marquee stars—like Josh Groban, now appearing in the Russian-inspired musical “Natasha, Pierre & The Great Comet of 1812” and Bette Midler in a revival of “Hello, Dolly!”—still soak up most of the applause. Sunday’s Tony awards will, as always, serve as a coronation for Broadway’s megawatt celebrities.

For a small niche of Broadway fans, however, social media is enabling them to indulge a quirky passion—watching fill-in performers. Standbys including Ms. Hurder often share news on social media about when they will be stepping into the lead, and Broadway Understudies, a fan group, provides updates on multiple web outlets. The group’s Twitter feed has more than 17,000 followers.