Nicknamed El Mudo (The Mute), Vázquez started his career at Belgrano, with whom he debuted in 2007 and then became a mainstay at the club, achieving interest from a number of Italian clubs (the first being Parma in 2010) as a consequence. In August 2011 he was heavily linked with a move to Palermo as a replacement for fellow Argentine Javier Pastore; the move, effective from January 2012, was subsequently confirmed on 1 September 2011 with a bid of €4.53m being paid by the Sicilians as a result. He officially joined the Sicilians' squad on 29 December 2011. He debuted in Palermo on 8 January against Napoli, and made a total 14 appearances with no goals (mostly as a substitute) .
On 24 August 2012 he moved on loan to Liga BBVA club Rayo Vallecano. He mostly played as a substitute, and failed to make a breakthrough in the first team squad, not being signed permanently by the club at the end of the season and therefore returning at Palermo. He was originally kept out of the squad list by head coach Gennaro Gattuso and therefore ineligible to play in any games for the first half of the 2013–14 Serie B season, but reincluded into the first team roster by new boss Giuseppe Iachini later in January 2014 as a replacement to on-loan youngster Valerio Verre.