Grisham is a native of Fort Worth, Texas and played his high school ball at Richland HS in North Richland Hills. Initially, the stocky outfielder was committed to stay in-state and play his college baseball at Texas Tech University, but the majors came calling after he was selected with the 15th-overall pick in the 2015 MLB Draft by the Milwaukee Brewers.
After spending time with numerous minor league teams, including the Class AA Biloxi Shuckers. On August 1, 2019, Grisham was promoted to the majors and made his debut on the same day against the Oakland Athletics.