Wright's interest in acting was inspired when she saw Akeelah and the Bee and found the lead, Akeelah Anderson (Keke Palmer) to be a positive role model. Determined to act, she sent emails to different agents about her acting experience until she was picked for small parts. She starred in My Brother the Devil, where she was recognized by Screen International as one of its 2012 Stars of Tomorrow. Michael Caton-Jones cast her in Urban Hymn which garnered her attention. She has played in numerous episodes of British TV series (see table below).

The British Academy of Film and Television Arts (BAFTA) named Wright among the 2015 group of BAFTA Breakthrough Brits. Wright is best known as Shuri in Marvel Cinematic Universe's Black Panther and Avengers: Infinity War (both 2018).

Wright has been vocal about her struggles with depression. In 2018 she told Vanity Fair, "I was in the dark going through so many bad things, when the world didn’t know about Shuri and Letitia and whatever is happening now." Wright took a break from acting until 2015 after she encountered Christianity at a London actors' Bible study. On the UK talk show This Morning, she explained, "I needed to take a break from acting, because I really idolized it. So I came off from it and I went on a journey to discover my relationship with God, and I became a Christian."

Wright co-starred in Black Panther playing the role of Shuri, King T’Challa’s sister and Princess of Wakanda. The film was released worldwide in February 2018 and co-starred Michael B. Jordan, Lupita Nyong'o, Danai Gurira, and Chadwick Boseman. She will reprise the role in the next Marvel Cinematic Universe film, Avengers: Infinity War, scheduled for release in April 2018.