On 13 December 2019, 15-year-old Lambert made his professional debut for
HIFK of the
SM-liiga, the top league in Finland. Following the season, in which he scored 38 points in 42 games with HIFK's
U20 club and two points in four games with their senior club, Lambert exercised a contractual option to leave the organization. On 8 May 2020, Lambert signed a two-year contract with Finnish club
JYP Jyväskylä. He would score 15 points in 46 games in the
2020–21 season, leading all under-18 players. Entering the
2021–22 season, Lambert was considered one of the top prospects for the upcoming
2022 NHL entry draft, ranked as high as second overall by some scouts. After a disastrous start to the season that saw him score only six points in 24 games, Lambert's contract was terminated by mutual agreement and he signed with his hometown club, the
Lahti Pelicans. In Lahti, his production stalled even further, to just four points in 25 games. Following this tumultuous and disappointing season, he fell in the draft until he was selected by the
Winnipeg Jets in the first round, 30th overall. In June 2022, despite never playing a game in the league and all signs pointing to him playing at the professional level upon arriving in North America, the
Seattle Thunderbirds acquired Lambert's
Western Hockey League (WHL) rights from the
Saskatoon Blades in exchange for draft picks. Lambert attended Jets training camp prior to the
2022–23 season. On 12 October 2022, he was signed to a three-year entry-level contract and assigned to the Jets'
American Hockey League (AHL) affiliate, the
Manitoba Moose. After 14 games with the Moose in which he scored only three points, the Jets assigned Lambert to the Thunderbirds. With the Thunderbirds, Lambert scored 48 points in 26 games, as well as 26 points in 17 playoff games en route to winning the
2023 Ed Chynoweth Cup as WHL champions. The Thunderbirds then reached the final of the
2023 Memorial Cup, where they were defeated by the
Quebec Remparts. Returning to the Moose for the
2023–24 season, Lambert was named to the 2024 AHL All-Star Game. With multiple Jets regulars not playing the final game of the season to rest ahead of the playoffs, Lambert was recalled to the NHL. He finished the AHL season with 20 goals and 52 points, good for second in rookie point scoring and earning a spot on the AHL All-Rookie Team. In his NHL debut on 18 April 2024, Lambert recorded an assist in a 4–2 victory over the
Vancouver Canucks. ==International play==