Having joined Club Guaraní's youth setup in 2020, Vásquez made his senior league debut in February 2021 against Club Olimpia.[3] He saved a penalty in the final match of the 2021 Clausura season against Sportivo Luqueño.[4] In 2022, Vásquez became the undisputed starting goalkeeper for Club Guaraní following the departure of the previously established number one Gaspar Servio on loan to Rosario Central, playing matches in both the league and the Copa Libertadores. His form was such that he was mentioned to be in the reckoning for a call-up to the Colombia national football team in January 2023.[5] Part of Vázquez's good form was saving multiple penalties during a shootout to knock Club Libertad out of the semi-finals of the Copa Paraguay.[6]
AC Milan
On 3 January 2023, Vásquez signed for Serie A club AC Milan on a three-and-a-half-year contract.[7]
Loan to Sheffield Wednesday
On 5 August 2023, Vásquez moved to England and joined recently promoted EFL Championship club Sheffield Wednesday on loan until 30 June 2024.[8] He made his debut for the club on 8 August 2023, in the EFL Cup against Stockport County. The match was tied and went to a penalty shoot-out where Vásquez saved two penalties to help secure a win for his side.[9][10] On 18 January 2024, he was recalled from his loan spell.[11]
Loan to Ascoli
On 19 January 2024, Vásquez was loaned to Serie B club Ascoli for the rest of the season.[12]
Loan to Empoli
On 18 July 2024, he was loaned to Serie A club Empoli, with an option to make the move permanent at the end of the 2024–25 season.[13]