Chichizola started his career with Club Sanjustino de San Justo.[2] In 2008, he signed for River Plate, where he made his professional debut on 13 February 2011 in the first round of the Clausura 2011 tournament in a 0–0 draw against Tigre.[3] He continued to play as River's first choice goalkeeper keeping clean sheets in all of his first four games for the club. On 13 March 2011, he conceded his first goal in a game against Vélez Sársfield after miskicking the ball straight to the opposition striker Santiago Silva.[4]
In the following game against Arsenal de Sarandí he was dropped from the first team after the usual first team goalkeeper Juan Pablo Carrizo returned to fitness.
On 29 June 2017, Chichizola signed for Spanish La Liga club UD Las Palmas.[5] He made his debut for the club on 18 August, making several key stops in a 0–1 away loss against Valencia CF.[6]
On 4 July 2018, Chichizola signed a three-year contract with Getafe CF.[7] Mainly a backup option during the most of his spell, he became a third-choice ahead of the 2020–21 season, after the return of Rubén Yáñez; on 5 January 2021, he terminated his link with the club.[8]