Rose was born and raised in Cincinnati, Ohio with three siblings, John, Nora and Mary. She played competitive soccer in her hometown, initially, with Sycamore United Soccer Club and later with the Cincinnati United Premier Soccer Club.
At Mount Notre Dame High School, Lavelle was a four-year started and was named Cincinnati's Player of the Year by the Cincinnati Enquirer her senior year. That same year, she netted 15 goals and scored a total of 38 points for her team. At the end of her high school career, Lavelle led her team in number of goals scored, totaling 57, and was named NSCAA All-Region her junior and senior years. She earned first-team all-state honors and was named the Greater Cincinnati/Northern Kentucky Sports Women of the Year award in 2013.
Accepted to the University of Wisconsin–Madison in 2013, Lavelle was a starting player in her first 19 matches in the Big Ten Conference. Her freshman year, she led the team with 73 shots and was second on the team with six goals. As a junior, she was named First-Team All-American by NSCAA.
On November 24, 2015, Lavelle was called to train with the United States Women's National Soccer Team. She was one of eight players who joined the team's Victory Tour following the 2015 Women's World Cup. She made her first appearance for the United States on March 4, 2017, during a match against England at the 2017 She Believes Cup. She was named Player of the Match following the team's 1–0 loss. Despite some setbacks due to injuries, Rose has been a consistent starter in midfield and was a key player during the team's victory at 2018 CONCACAF Women's Championship, which earned them a spot in the 2019 Women's World Cup.
In May 2019 Lavelle was named to the US Women's National Team roster for the 2019 World Cup in France. She started six games, scoring three goals including the second goal during the US's 2-0 win over the Netherlands in the Championship Match. As a result of her outstanding performance, she was named the winner of the Bronze Ball as the tournament's 3rd place player.