While in the Bristol Bears academy he was dual-registered with Clifton, and played mainly for the amateur club.[3]
He was offered the chance to go to New Zealand in the summer of 2016, when he played for Sumner in the Canterbury area.[3]
He joined the Bristol Bears academy in 2016. He played for Bristol in the Premiership Sevens.[3]
He played for Hartpury University while getting his degree in Strength and Conditioning, first of all on a loan deal, then from 2017 in the English Championship.[4][5]
In 2020, he signed for Ayrshire Bulls in the Super 6.[6]
He was training with Glasgow Warriors in 2022.[7][8] Head Coach Franco Smith said: "We brought Eli in after the end of the Super6 Championship in late October and he’s trained really well with us. He played for our A team in a training match against Scotland under-20s last month, scoring three tries, and it’s an area of the team that we want to broaden the number of players available to us."[9]
He made his competitive debut for Glasgow Warriors on 25 February 2023, in the match against the Lions in the United Rugby Championship, scoring a try in the match. Caven earned the Glasgow Warrior No. 351.[10]
He moved to play for Cambridge for the 2023-24 season.[11]
He has a Bachelor of Science in Strength and Conditioning. He is also a BWL Olympic Weightlifting Coach. He was a Strength and Conditioning Coach for Ayrshire Bulls till moving in 2023.[12][13]