Mueller was born and raised in Evanston, Illinois, the daughter of actors Jill (née Shellabarger) and Roger Mueller. Her two brothers Andrew and Matt along with sister Abby are all actors known for their work in the Chicago theatre community. Mueller is Lutheran. She graduated from Evanston Township High School in 2001 and from Syracuse University in 2005.
After graduation, Mueller returned to Chicago and acted for the Chicago Shakespeare Theater, traveling in 2006 to England to perform at the Swan Theatre in Stratford-upon-Avon as Lady Mortimer in Henry IV. She followed this with various roles in the Chicago area including Lizzie Fields in Baby, Once Upon a Mattress as the Lady-In Waiting, and Esther in Meet Me in St. Louis. At the Marriott Theatre in Lincolnshire, Illinois, she performed roles in Shout! The Mod Musical, Shenandoah, Miss Adelaide in Guys and Dolls, and Tzeitel in Fiddler on the Roof. In 2009 Mueller performed in two musicals at the Goodman Theatre, Animal Crackers as Grace/Mary and A Christmas Carol as Belle/Catherine.
In 2011, she was named Actor of the Year by the Chicago Tribune and also was nominated for two Joseph Jefferson Awards, winning Best Lead Actress in a Musical for her performance as Amalia Balash in She Loves Me.
Mueller made her Broadway debut in a revised revival of On a Clear Day You Can See Forever as Melinda Wells, a role originated by Barbara Harris. The musical opened in December 2011. Mueller received critical praise for her performance, with one reviewer commenting "...Her voice contains notes of Garland, but she’s no diva — this is a star of supreme self-possession". For this role, Mueller was nominated for the Tony Award for Best Featured Actress in a Musical.
Mueller next was cast in the role of Helena Landless / Miss Janet Conover in Roundabout Theatre Company's 2012 revival of The Mystery of Edwin Drood for which she was nominated for a Drama Desk Award. Mueller portrayed Cinderella in a 2012 Public Theater production of Stephen Sondheim's Into the Woods at the Delacorte Theater. She then took over the role of Billie Bendix in Nice Work If You Can Get It, from March 29, 2013 until the show closed on June 15, 2013. Also in 2013, she once again took on the role of Carrie Pipperidge in Carousel at Lincoln Center in a staged concert production backed by the New York Philharmonic - a special that was filmed and aired on PBS' Live from Lincoln Center.
Mueller appeared in the musical Beautiful: The Carole King Musical, which opened on Broadway on January 12, 2014. She originated the title role of Carole King. She had performed this role in the try-out in San Francisco in 2013. She won the Drama Desk Award and Tony Award for Best Lead Actress in a Musical. She left the Broadway production on March 6, 2015. On February 8, 2015, Mueller won a Grammy award for Best Musical Theater Album (shared with Carole King (music and lyrics), and producers Jason Howland, Steve Sidwell & Billy Jay Stein).
On April 27, 2015, Mueller was the recipient of the prestigious 2015 Sarah Siddons Society Award at the Marriott Theatre in Lincolnshire, Illinois. The Sarah Siddons Society, founded in 1952 by prominent Chicago theater patrons determined to promote excellence in the art, presents its annual award (named after the 18th century Welsh actress), to an actor of outstanding achievement. Previous winners have included such legends as Audra McDonald, Barbara Cook, Helen Hayes, Geraldine Page, Carol Channing, Bette Davis, Angela Lansbury, Lauren Bacall, Julie Andrews, Chita Rivera, Patti LuPone and many more.
Mueller reprised her role as Jenna in the musical adaptation of the film Waitress on Broadway, having previously played the role in a workshop of the musical in late 2014. With music and lyrics by Sara Bareilles and direction by Diane Paulus, the musical premiered at the American Repertory Theater in August 2015.
Waitress opened at the Brooks Atkinson Theatre on April 24, 2016. Mueller was nominated for a Tony Award and a Drama Desk Award, among others, for her performance. Mueller also received critical praise for her performance, Charles Isherwood for the New York Times noted: "But it is when Ms. Mueller tears into the musical’s climactic number, 'She Used to Be Mine,' that her talents are most fully and movingly harnessed... Ms. Mueller’s vibrant performance of this gentle but wrenching song, featuring a melody that soars and then recedes in waves, is the high point of the show — and for that matter a high point of the Broadway season. Suddenly, a pleasant and polished but weightless musical comedy rises to transporting heights, and sweeps up your heart along with it."
Mueller left the production on March 26, 2017.
In April 2017, shortly after her departure from Waitress, it was announced that Mueller will next portray the lead role of Julie Jordan in a Broadway revival of Carousel, which will open April 12, 2018 at the Imperial Theatre with previews beginning February 28.
It was later announced that Mueller will also be making her feature film debut in Steven Spielberg's upcoming historical drama The Post. Mueller will portray real life advice columnist Judith Martin (a.k.a. Miss Manners) alongside a cast that includes Meryl Streep, Tom Hanks, Sarah Paulson, and Carrie Coon. The film is scheduled for a limited release on December 22, 2017 with a wide release on January 12, 2018.