Club 

Lauren was spotted by scouts from Cardiff City. She credited her kickboxing training with helping improve her play, commenting "I could kick a ball a lot further than any of my teammates" and noting that "I might have had a higher pain threshold than everyone else." She was part of the club's under-16 side that won the Welsh section of the Tesco Cup in 2010, being named player of the tournament.

She progressed to the senior team at Cardiff and won the inaugural Welsh Premier Women's Football League title during the 2012–13 season after their decisive 5–2 victory over Wrexham in the final game of the season. Price was named the club's Player of the Year during their title winning season. Price was also named the Football Association of Wales (FAW) Club Player of the Year. In 2014, Price stepped away from football to concentrate on her boxing career ahead of the 2014 Commonwealth Games.

Price initially took up boxing as a teenager but became further involved in the sport after watching British fighter Nicola Adams win gold at the 2012 Summer Olympics in London. After competing in a single amateur about at 17, she entered the Women's European and Youth World Championships where she claimed a bronze medal. At the 2014 Commonwealth Games, she became the first Welsh woman to claim a boxing medal in the Commonwealth Games after defeating Kaye Scott in the quarter-finals of the women's middleweight division to guarantee at least a bronze. She met Ariane Fortin in the semi-final but suffered a split decision defeat.

Olympics

She claimed another bronze medal at the 2016 Women's European Amateur Boxing Championships. In 2018, she won gold at the 2018 Commonwealth Games, after defeating Caitlin Parker via split decision in the final of the event. In May 2019, Price was selected to compete at the 2019 European Games in Minsk, Belarus, winning the gold medal.

Price competed in the 2020 Summer Olympics in Tokyo, beating Dutch boxer Nouchka Fontijn in the middleweight semi-final. She went on to win gold against Li Qian of China, making her the first Welsh boxer of any gender to win an Olympic gold medal. Price was appointed Member of the Order of the British Empire (MBE) in the 2022 New Year Honours for services to boxing.

Price studied a Foundation Degree in Football Coaching and Development at the University of South Wales. Her partner is featherweight boxer and fellow southpaw Karriss Artingstall.