Yamamoto's paternal grandfather is Japanese, and his maternal grandmother is a native of Hawaii. He played three seasons for Spokane of the Western Hockey League after he was the No. 10 pick in the 2013 WHL Bantam Draft by his hometown Chiefs. In 2017-18, his third season in Spokane, Yamamoto had WHL highs of 42 goals, 57 assists and 99 points, and was named to the Western Conference Second All-Star Team.

The forward was chosen by the Edmonton Oilers in the first round (No. 22) of the 2017 NHL Draft and made his NHL debut in a 3-0 win against the Calgary Flames on Oct. 4, 2017. He was returned to Spokane after his ninth NHL game and played 17 games for the Oilers the following season, scoring his first NHL goal in a 3-2 overtime win against the Boston Bruins on Oct. 18, 2018.

Yamamoto signed a two-year, $6.2 million contract with the Oilers on Aug. 3, 2022. He scored the go-ahead goal with 3:03 remaining, and the Edmonton Oilers eliminated the Los Angeles Kings with a 5-4 win in Game 6 of the Western Conference First Round.

Yamamoto and Klim Kostin were traded by Edmonton to the Detroit Red Wings on June 29, 2023, for future considerations. On July 2, Yamamoto signed a one-year, $1.5 million contract with the Seattle Kraken.

Yamamoto signed a one-year, two-way contract with the Utah Hockey Club on Oct. 6, 2024, after taking part in training camp on a professional tryout contract.

Internationally, Yamamoto represented the United States at the 2014 World Under-17 Hockey Challenge, 2016 IIHF Under-18 World Championship and 2018 IIHF World Junior Championship, winning silver, bronze and bronze, respectively. He also had seven points (four goals, three assists) in four games at the 2015 Ivan Hlinka Memorial Tournament, tying for the tournament lead in goals and points.