The 18-year-old came on from the bench in Wales' 33-0 win against Russia in Wellington on Saturday.
Botham is the grandson of former England cricketer Sir Ian Botham and son of former Cardiff wing Liam.
Liam turned down an offer to play a trial for Wales during his career, saying he wanted to play for England.
James, who is part of the Cardiff Blues academy, has also represented Wales Under-18 and scored on his debut against Scotland in 2016.
Wales went on to lose to 28-5 to Canada in a shock defeat in Wellington before they were beaten 31-7 in their final Pool D game.
James Botham made his debut appearance for Wales Sevens in the World Sevens Series in New Zealand, committing him to Wales for life.