Born in Zaragoza, Aragon, Vallejo joined Real Zaragoza’s youth setup in 2007, aged 10. On 26 July 2013, after impressing in the Juvenil squad, he signed a new deal with the club.

On 7 March 2013, Vallejo appeared with the Spain under-16 team in a friendly with Hungary. He earned his first cap for the under-21s on 26 March 2015 at the age of 18, starting in the 2–0 win against Norway in Cartagena in another exhibition game.

On 23 August 2014, before even having appeared for the reserves, Vallejo made his professional debut, starting in a 0–0 draw at Recreativo de Huelva for the Segunda División. On 26 December, he further extended his contract until 2019.

On 31 July 2015, Vallejo signed a six-year contract with Real Madrid for a rumoured €6 million fee, being immediately loaned back to Zaragoza for one year. The following summer he moved abroad, after agreeing to a one-year loan deal with German club Eintracht Frankfurt.

Vallejo’s maiden appearance in the Bundesliga took place on 27 August 2016, as he came as a late substitute in a 1–0 home win over FC Schalke 04. He scored his first goal also from the bench, contributing to a 2–2 draw to RB Leipzig also at the Commerzbank-Arena in the last match of the season.

On 7 July 2017, Vallejo was unveiled as a Real Madrid player and member of the first team squad for the upcoming campaign. He was handed the number 3 shirt, previously worn by Pepe. His official debut took place on 26 October, when he started and was sent off in the last minute of a 2–0 away win against CF Fuenlabrada for the Copa del Rey. His maiden appearance in La Liga occurred ten days later, when he partnered Sergio Ramos in the 3–0 home defeat of UD Las Palmas.

Profiting from injury to Nacho and suspension to Ramos, Vallejo made his debut in the UEFA Champions League on 11 April 2018, playing the entire 1–3 home loss to Juventus F.C. for the second leg of the quarter-finals, which still qualified to the last-four 4–3 on aggregate. It was his only appearance, as Madrid won their third consecutive and 13th overall title in the tournament.

On 27 July 2019, Vallejo moved to English Premier League side Wolverhampton Wanderers on a season-long loan. He made his competitive debut on 15 August, in a 4–0 home win against FC Pyunik in the third qualifying round in the UEFA Europa League. He played his first Premier League match on 14 September, featuring the full 90 minutes in the 5–2 loss to Chelsea also at Molineux. He made one further league appearance, but was not used again except for an EFL Cup tie in a side consisting largely of reserve players.

At the start of the January transfer window, Wolves’ head coach Nuno Espírito Santo confirmed that Vallejo was likely to leave and added that the defender “had moments that he played, moments that he performed well, and some moments he didn’t perform well … Clearly it didn’t work out”. On 24 January 2020, he was loaned to Granada CF of the Spanish top level until June; on 18 August, the move was extended for another year.