Seattle Sounders FC
Vargas played in the academy of Seattle Sounders FC for two seasons before signing a professional contract with Sounders FC reserve side Tacoma Defiance on May 7, 2021. Two days later, he made his senior debut for Tacoma Defiance in the USL Championship against LA Galaxy II. Vargas started and played 66 minutes as the match ended in a 1–1 draw.
He made his MLS debut for Seattle on July 22, 2021, as one of several Tacoma Defiance players called up to the Sounders for a match against Austin FC to replace injured players. At 15 years and 351 days old, Vargas was the third-youngest player in MLS history. He signed a formal Homegrown Player contract with the Sounders in December and became a starter for the team in the first matches of the 2022 season, playing in two CONCACAF Champions League fixtures. Head coach Brian Schmetzer described Vargas' play style as similar to that of Cristian Roldan's rookie season, adding he was "going to be a tremendous player for our club". He was an early substitute in the second leg of the 2022 CONCACAF Champions League final, which the Sounders won 3–0 at home.
Vargas made 13 appearances through late June, when a stress facture in his back prevented him from continuing with the team. He was also unable to join the under-20 national team in the CONCACAF U-20 Championship. Vargas returned for the 2023 season but played in few matches, including only 24 minutes in the MLS Cup playoffs. He regained the starting position in midfield for the 2024 season and scored his first career MLS goal on April 30, 2024, against the Philadelphia Union, in a match that was originally started a month earlier but postponed due to heavy rain. Vargas played over 3,000 minutes for the Sounders in the 2024 season and was ranked seventh in the MLS 22 Under 22 list at the end of the regular season. He was suspended for the first match of the 2024 playoffs after receiving two yellow card in a match against the Portland Timbers.
During the 2025 regular season, Vargas started in 26 matches and finished second on the team in minutes played. He was a starter for Seattle's three matches in the 2025 FIFA Club World Cup group stage and played for the team's entire 2025 Leagues Cup run, where the Sounders won the title. Vargas was named to the roster for the 2025 MLS All-Star Game and played in the match; he was also ranked first in the 2025 MLS 22 Under 22 list. Vargas was also a finalist for the 2025 MLS Young Player of the Year Award. In his five seasons with the Sounders, Vargas scored 8 goals in 130 appearances across all competitions.
Atlético Madrid
On February 2, 2026, La Liga club Atlético Madrid announced the signing of Vargas on a four-and-a-half-year contract in the closing hours of the transfer window. He had rejected an earlier offer from Club América of Liga MX to pursue a career in the European leagues. Vargas previously stated that Atlético Madrid had been his favorite team as a child; he also played against them in the 2025 FIFA Club World Cup. He made his debut for Atlético as a substitute in the 79th minute of a Copa del Rey quarterfinal match against Real Betis. He recorded his first goal contribution for Atlético Madrid on May 2nd 2026, providing an assist in a 2–0 victory over Valencia.