Raised in Studio City, California in the Los Angeles area, Alyssa attended The Wesley School in North Hollywood.

[16] At the time, this trade made Thompson "essentially the most expensive player in league history," which was a record previously set by Alex Morgan.

[17] While still attending her senior year of high school at Westlake,[18][19] Thompson scored five minutes into her professional debut with Angel City FC, in a friendly match against Club América of Liga MX Femenil.

[28] Thompson converted her first penalty kick for Angel City on August 1, 2024, in a shootout against San Diego Wave FC during the 2024 NWSL x Liga MX Femenil Summer Cup to ensure a spot in the knockout stage of the tournament.

[29] Thompson also convereted her first non-shootout penalty for Angel City a few weeks later in a friendly against Liga MX Femenil team FC Juárez on August 18, 2024, her first goal of the 2024 season.

[30] Thompson scored her first career brace and her first regular-season goals of the 2024 season on August 24, 2024, in an away match against San Diego Wave FC to secure a 1–2 victory.

[34] On January 29, 2025, ahead of the 2025 season, Angel City announced they had signed a contract extension with Thompson, keeping her and her sister at the club until 2028.

In 2022, she helped lead the under-20 national team to win gold at the 2022 CONCACAF Women's U-20 Championship, playing in five matches and scoring three goals.

[39][40][12] On October 7, 2022, she made her senior national team debut at Wembley Stadium during a friendly against England in front of 90,000 fans.

[44] Thompson recorded her first assist for the senior national team on October 29, 2023, setting up Jaedyn Shaw's goal in a 3–0 win in a friendly against Colombia.

[8][54] In May 2022, Thompson and her sister Gisele signed a name, image, and likeness deal with Nike at ages 17 and 16 respectively, becoming the first high school athletes to do so.