Socha was born in Nashua, New Hampshire, where she grew up, the daughter of Joan Storey and Thad Socha. As an only child, she performed with her parents in local community theater shows. She attended Nashua High South graduating in 2008. Socha began to act as a child, joining the Peacock Players of Nashua, where she played Amneris in the musical Aida in 2006. She gained professional experience with the American Stage Festival and The Lyric Stage.
After completing her junior year of high school, Socha made her Broadway debut in Spring Awakening on July 24, 2007, as an ensemble member and understudy for Wendla, Thea, Anna and Martha (replacing Krysta Rodriguez). On May 20, 2008, Socha took over as Wendla (replacing Lea Michele). She received positive notices in the Daily News and The New York Times. She played out the remainder of the run, and also graduated from Nashua High School in 2008 after completing a home study program. Her next Broadway appearance was the short-lived revival of Brighton Beach Memoirs, which opened on October 25, 2009.
In September 2010, Socha was at the Yale Repertory Theater in "We Have Always Lived In The Castle," a new musical based on the Shirley Jackson novel of the same title. In April 2011, Socha opened in The Dream of the Burning Boy, at the Roundabout Theater Company's Black Box Theater.
On July 14, 2011, she opened as Daisy Fenton in Death Takes a Holiday, at the Laura Pels theater.
On November 28, 2011, Socha made her solo concert debut at Duplex Cabaret in New York City with a show entitled Home.
On October 17, 2012, Socha played Medium Alison in a workshop of the musical Fun Home, at the Public Theater in New York.
From Sept. 30 – Dec. 1, 2013, Socha originated the role of Medium Alison in the official off-Broadway premiere of Fun Home at The Public Theater in New York.
Her television work includes the character Mia in the Showtime series The Big C, single appearances on White Collar, A Gifted Man, Made in Jersey, Law And Order: SVU, and a recurring role in the FX series Damages. She currently features in a leading role in the Amazon Comedy TV series Red Oaks.