Azócar signed with the Detroit Tigers as an international free agent on October 21, 2012 and would make his professional debut in 2013 with the VSL Tigers (the Venezuelan Summer League). He would then join the Low–A Connecticut Tigers in 2015, before quickly being sent to the rookie–level Gulf Coast League Tigers, where he was named a 2015 Post Season All-Star. Azócar hit .325 with 29 RBI, 10 doubles, and five triples. His play warranted a call–up for the 2016 season to the Single–A West Michigan Whitecaps, where he spent the entire year. Azócar made another jump in play up to the High–A Lakeland Flying Tigers for 2017, however despite seeing his first professional home runs (he would hit three), his average would drop from .281 in 2016 to .220 in 2017.
Azócar would spend 2018 in both West Michigan and Lakeland. He would start 2019 with the Double–A Erie SeaWolves. As of May 11, 2019, he possessed the third-highest batting average in the Eastern League with a .336 clip.[1] Azócar's average would ultimately take a downtick, but it would be sitting at .293 when, on June 24, he was named to the Eastern League All-Star Game for the Western Division.[2] At time of the honoring, he led the league in hits with 136, was tied for third-highest batting average in the league (.284), had 39 multi-hit games, and also had ten home runs and 56 RBI.[3] Azócar would ultimately earn the dubious distinction of leading the minor leagues in 2019 by grounding into 22 double plays.[4] However, he would be named the Eastern League Rookie of the Year for 2019, becoming the first SeaWolves' player to earn the honor. Azócar was re-signed by the Tigers organization after becoming a minor league free agent on November 7, 2019.[5] He did not play in a game in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[6] Azócar became a free agent on November 2, 2020.[7]
San Diego Padres
On November 18, 2020, Azócar signed a minor league contract with the San Diego Padres. He split the 2021 campaign between the Double–A San Antonio Missions and Triple–A El Paso Chihuahuas. In 128 games split between the two affiliates, Azócar slashed .281/.341/.438 with nine home runs, 70 RBI, and 32 stolen bases.[8]
On April 7, 2022, Azócar had his contract selected to the major league roster and made his MLB debut that night as a pinch runner and left fielder. In his first at-bat two days later, he had his first major-league hit, a single, and drove in his first run.[9] That year, he appeared in two games in the NLDS and one in the NLCS for the Padres without coming up to bat. He served as a pinch runner and a replacement left fielder. He did steal a base in Game 2 of the NLDS. On September 15, 2023, Azócar hit his first MLB home run, driving in 3 runs during the top of the ninth inning against the Oakland Athletics.[10]
Azócar played in 61 games for the Padres in 2024, slashing .219/.269/.247 with two RBI and five stolen bases. Azócar was designated for assignment by the Padres on September 2, 2024.[11][12]
New York Mets
On September 5, 2024, Azócar was claimed off of waivers by the New York Mets. However, he was then optioned to the Triple–A Syracuse Mets.[13]