Grisham was born and raised in the Fort Worth, Texas area, where he attended Richland High School and emerged as a top MLB draft prospect.

Grisham spent 2016 with the Wisconsin Timber Rattlers where he batted .231 with two home runs and 24 RBIs in only 59 games due to hamstring injuries.

[7] In 2017, Grisham played for the Carolina Mudcats, posting a .223 batting average with eight home runs, 45 RBIs and 37 stolen bases in 133 games.

[10] On November 27, 2019, the Brewers traded Grisham, Zach Davies, and cash considerations or a player to be named later to the San Diego Padres in exchange for Luis Urías and Eric Lauer.

[11] On August 22, 2020, Grisham had his first career multi-home run game, hitting homers in the first and second innings against Brandon Bielak of the Houston Astros.

He also led NL outfielders with 134 putouts and won the National League Gold Glove Award as a center fielder.

[13] Grisham began the 2021 season on the IL with a hamstring strain he suffered early in spring training, but he rejoined the team on April 9.

[19] On December 6, 2023, the Padres traded Grisham and Juan Soto to the New York Yankees for Michael King, Drew Thorpe, Jhony Brito, Randy Vásquez, and Kyle Higashioka.