Swallow started her career in theatre, where she performed in various productions, including the Broadway musical High Fidelity, King Lear, The Taming of the Shrew, A Midsummer Night's Dream, Cat on a Hot Tin Roof at the Guthrie Theater, Much Ado About Nothing for Shakespeare in the Park, and the world-premiere of off-Broadway shows Romantic Poetry and Measure for Pleasure.

She is best known for her role as The Armorer in the Star Wars series The Mandalorian. She is also known for her roles as Kim Fischer in The Mentalist and as Amara / The Darkness in the 11th season of Supernatural. She also had a minor role as Emily in the video game The Last of Us Part II.