Vázquez attended the Puerto Rico Baseball Academy and High School, and was selected by the Boston Red Sox in the ninth round of the 2008 MLB Draft.

[1] His first professional team was the rookie league Gulf Coast Red Sox in 2008 and part of 2009, where he batted 21-for-94 (.223) in 31 games.

In 2012, he batted a combined .254 with seven home runs and 46 RBIs in a total of 101 games for the Class A-Advanced Salem Red Sox and the Double-A Portland Sea Dogs, compiling a .986 fielding percentage while throwing out a 40% of potential base stealers (49-for-122).

In 2014, Vázquez had a slash line of .275/.331/.715 in his first 62 games for Triple-A Pawtucket, and was named to the International League All-Star team.

[3] Vázquez made his MLB debut on July 9, starting at catcher against the Chicago White Sox; he was hitless in three at bats, while defensively tagging out a runner at the plate when Alejandro De Aza tried to score from first base on a double by Tyler Flowers.

On April 2, 2015, Vázquez underwent Tommy John surgery and was placed on the 60-day disabled list;[7] he missed the entire season.

During the 2017 Red Sox regular season, Vázquez batted .290 with five home runs and 32 RBIs; he also had seven stolen bases.

On June 21, Vázquez hit a walk-off two-run home run in the 10th to cap a comeback against the Toronto Blue Jays.

[17] On August 1, 2022, the Red Sox traded Vázquez to the Houston Astros for Wilyer Abreu and Enmanuel Valdez.

[24] In the seventh inning of the regular season finale on October 5 versus the Philadelphia Phillies, Vázquez homered for his first as an Astro.

He also caught the Astros' combined no-hitter in Game 4 of the World Series, hurled by Cristian Javier, Bryan Abreu, Rafael Montero, and Ryan Pressly.

[31] Vázquez credits his development as a catcher in part to the Molina brothers (Yadier, Jose, and Bengie), who have all played in MLB.