He made his collegiate debut on September 5, 2009, against Santa Clara, making four saves in a 3–2 defeat at the UC Irvine College Classic.

"[18] Although Stuver spent the majority of the season training with Columbus Crew, he made his first appearance on a MLS gameday roster with Chivas USA.

Stuver spent the entirety of the 2014 season on loan in USL Pro, with Dayton Dutch Lions and Wilmington Hammerheads FC.

[25] Stuver was promoted to the backup role for Columbus Crew SC to begin the 2015 season, jumping Matt Lampson on the depth chart.

He made his debut for Crew SC in a friendly against Valencia CF on May 27, conceding a goal to Álvaro Negredo in the 33rd minute of a 1–0 defeat for Columbus.

[31] After starting keeper Steve Clark went down injured at the end of the season, Stuver made his MLS debut against the New York Red Bulls on October 16.

[32] Stuver also started the season finale away to New York City FC on October 23, but made just two saves in a 4–1 defeat for Crew SC.

He missed time early in the season with a right elbow strain, although Steffen played every minute of the seven games Stuver was unavailable.

Stuver made two saves, but conceded a 64th-minute goal to Djiby Fall as Crew SC were defeated 1–0 in the first-ever Hell is Real Derby.

[36] Stuver again went the full 90 minutes in a friendly for Columbus, keeping a clean sheet in a 1–0 victory over Eintracht Frankfurt on July 17.

He made his league debut for the Dutch Lions on May 25 against Sacramento Republic FC, conceding goals to Dakota Collins and Octavio Guzmán in a 2–1 defeat.

[43] After undergoing ankle surgery in June, Columbus announced on August 4, 2014, that Stuver would again be sent on loan, this time to Wilmington Hammerheads FC of USL Pro.

[44] He made his debut for Wilmington on August 9 against his former club, Dayton Dutch Lions, conceding goals to Joe Broekhuizen and Karamba Janneh in a 2–1 defeat.

He helped Wilmington make it to the first round of the USL Pro playoffs as the seventh seed, where they were defeated by eventual league champions Sacramento Republic FC, 4–1.

[47] On December 14, 2017, Stuver was traded by Columbus to New York City FC in exchange for a fourth-round pick in the 2019 MLS SuperDraft.

[52] In Austin's first-ever playoff game, Stuver saved two out of the three unsuccessful Real Salt Lake penalties in a 3–1 shootout win (2–2 at 120 minutes).

While still playing professionally, Stuver started coaching the goalkeepers for the Ohio Wesleyan Battling Bishops men's soccer program in 2015.