He grew up on Mercer Island, Washington, and joined the Sounders youth academy before playing college soccer at Stanford University.

Morris won the Hermann Trophy in his senior season at Stanford and was called up to the United States national team, scoring in a friendly against Mexico.

He signed for Seattle's senior team in 2016 as a Homegrown Player, winning an MLS Cup in his rookie season and a second title in 2019.

Morris was loaned to EFL Championship side Swansea City in early 2021, but returned after four matches due to an ACL tear.

Morris, from Mercer Island, Washington, began his youth career with Eastside FC, where he played from 2004 to 2012, from U11 to U17, with the Eastside FC B94 Red team, coached by Dan Strom, and helped the team to six of its seven Washington State titles as well as two third-place finishes at the US Youth Soccer National Championships in 2011 and 2012: he was named to the Best XI in 2011, and was the Golden Ball winner in 2012.

Later in January, Bremen extended Morris' trial who then played in a friendly match against Inter Baku PIK and recorded an assist.

It was widely speculated that Morris would sign with Seattle Sounders FC, the club for which his father works, and also holds his amateur rights.

[22] On January 21, Morris signed with Seattle Sounders FC, being given MLS's highest-ever Homegrown Player contract worth roughly $250,000 a year.

[33] On February 22, 2018, while playing in El Salvador against Santa Tecla in the Sounders' first match of the 2018 CONCACAF Champions League, Morris collapsed from a non-contact injury in the 85th minute and was later diagnosed with a torn anterior cruciate ligament (ACL).

[37] Morris continued playing during the COVID-19 pandemic and the MLS is Back Tournament despite an elevated risk due to his diabetes, citing his confidence in the club's precautionary measures.

[43] Morris underwent a successful knee surgery in March 2021 that was performed by Dr. Bert Mandelbaum, who had previously operated on him following his right ACL tear in 2018.

[43] He underwent rehabilitation with the Sounders for eight months and enrolled in Stanford University's online classes to finish his degree in science technology and society.

[47] He made an additional regular season appearance as a substitute and started for the Sounders in a playoffs loss to Real Salt Lake.

[51] In the fifth week of the 2023 regular season, Morris earned his second career hat-trick with the Sounders, scoring four goals in a 4–1 away win against Sporting Kansas City.

[61] He was named to the U.S. squad for the 2022 FIFA World Cup alongside teammate Cristian Roldan, a close friend and roommates on away trips for both teams.