Grant is the granddaughter of James Grant, a Scottish golf professional who emigrated from Inverness in Scotland to Helsingborg in Sweden.[3] The Grant family has had a fair bit of success on Catriona Matthew's home track, North Berwick Golf Club. James won the Scottish Boys Championship on the course, and Linn won the British Amateur Stroke Play Championship there, 49 years later. James died when Linn was 6 years old.[1] Her father John played on the Swedish Golf Tour and has seven wins on the Swedish Senior Tour.[4][5]
In 2018, Grant won the qualification for the 2018 Women's British Open at St Annes Old Links Golf Club with a round of 62 (−10), however she missed the cut in the Women's Open at Royal Lytham & St Annes Golf Club after rounds of 78 and 72. She also won the sectional qualification for the 2018 U.S. Women's Open at Buckinghamshire Golf Club with rounds of 67 and 70, seven strokes ahead of the runner-up Catriona Matthew and nine ahead of the third qualifier Mel Reid. In the U.S. Open at Shoal Creek Club, Grant was tied for fourth and best amateur after opening rounds of 69 and 72, and finished tied for 57th after final rounds of 78 and 81.[11][12]
In June 2022, Grant became the first woman to win on the European Tour, at the time named the DP World Tour, by winning the Volvo Car Scandinavian Mixed at Halmstad Golf Club in Sweden, a tournament with a field mixed of 78 women and 78 men, playing from different tees for the same title and the same prize money, but divided when counting for the women on the Ladies European Tour and for the men on the European Tour. Grant went into the final round with a two shot lead, and went on to score an 8-under-par 64 in the last round, and to win by nine strokes ahead of the nearest men and 14 strokes ahead of the nearest women.[21]
After finishing runner-up in 2020 and 2021, Grant won the 2022 Skaftö Open by one stroke ahead of Lisa Pettersson.
Grant played only six events in her rookie season on the LPGA Tour in 2022, garnering four top-10 finishes in the process. Although accumulating enough points to qualify for the CME Group Tour Championship, Grant did not participate in the tour's season-ending event due to her COVID vaccination status. Grant said through a statement via her agent, "with travel restrictions to enter the U.S. still remaining, it is still not an option for me to play LPGA events in the U.S. This is the sole reason I am not playing the CME."
In May 2023, travel restrictions to enter the U.S. were released, why Grant started playing LPGA Tour events in the U.S. In her fourth start on U.S. soil since the release, she captured her first LPGA Tour win in July 2023 at the Dana Open. It was the 120th Swedish LPGA Tour win and she became the 14th Swedish LPGA Tour winner. On the same day as her triumph, a Swedish player won on the men's tour, Vincent Norrman at the Barbasol Championship on the PGA Tour.[22]
In June 2024, Grant won on the Volvo Car Scandinavian Mixed for a second time, after starting the final day 11 shots behind 54-hole leader Sebastian Söderberg, achieving the biggest comeback ever in a golf tournament.[23]
In 2024, she was awarded honorary member of the PGA of Sweden.[24]