During her freshman year of college, in 2008, Glenn was cast as Thea in the first U.S. national tour of Steven Sater’s and Duncan Sheik’s rock musical Spring Awakening. In 2013, she booked her breakthrough role as Litchfield Penitentiary inmate Brook Soso in the Netflix comedy-drama series Orange Is the New Black, for which she received Screen Actors Guild Awards in 2014, 2015, and 2016 for Outstanding Performance by an Ensemble in a Comedy Series.
In 2014, she appeared in the Lena Dunham-directed music video “I Wanna Get Better” for Jack Antonoff’s solo project Bleachers.
Glenn played the supporting role of Liv Kurosawa in the drama-thriller film Nerve (2016), directed by Henry Joost and Ariel Schulman and based on the young adult novel of the same name. In 2016, she was cast as Dawn Pinkett in the Broadway transfer production of Sara Bareilles’s and Jessie Nelson’s musical Waitress. Preview performances began on March 25, 2016, at the Brooks Atkinson Theatre, and the show officially opened on April 24.
In 2018, Glenn originated the role of Peni Parker in Spider-Man: Into The Spiderverse.
In 2018, Glenn began starring as Harlow in the comedy Web television series Liza on Demand, with Travis Coles and creator Liza Koshy. In 2019, she was a guest presenter with Nev Schulman on the MTV show Catfish.
In 2021, Glenn provided the voice of Izzy Moonbow in the Netflix animated film My Little Pony: A New Generation.
In 2023, Glenn lended her voice to Kiff Chatterley, the eponymous protagonist of the Disney Channel animated original series Kiff.