Veglio signed his first professional contract in July 2018 with Montevideo Wanderers, effectively ending his seven-year youth career with the club.[2] He made his professional debut for the club on 26 February 2019, coming on as an 82nd-minute substitute for Francisco Ginella in a 4–1 victory over Nacional.[3][4] After breaking into the first team as a frequent starter the following season, Veglio suffered a season-ending knee injury in November 2020.[5] He would make his return to first team football at the season-opening 2021 Supercopa Uruguaya.[6]