Is mounted archery in the Olympics?

No. The only archery event in the Olympics is 'Target Archery'.

But it is unlikely to ever be added because of an animal being involved & the danger to said animal.

https://olympics.com/en/featured-news/archery-olympics-rules-disciplines-equipment-categories-bow-arrow

Upon reading this question and its subsequent answer, I wondered whether mounted archery had ever been part of the Olympic Games.

Archery was a sport in 1900, 1904, 1908 and 1920, Olympiads II (Paris), III (St. Louis), IV (London) and VII (Antwerp), but not as a mounted discipline. It was then dropped, owing to the number of local variations in the sport. World Archery (formed in 1931) finally managed to reinstate it at Munich (1972) using recurve bows. Team events started in 1988 at Seoul with a mixed team event held at Tokyo in 2020/21.

Many feel that both the Summer and Winter Olympic Games have too many events or associated disciplines. Rather than commendable animal safety, this is possibly the reason that mounted archery is unlikely to come an Olympic Sport.

Reference: https://olympics.com/en/featured-news/archery-olympics-rules-disciplines-equipment-categories-bow-arrow