A spokesperson for Buckingham Palace previously said King Charles III and other members of the royal family had been present at the arena at the time the incident took place.
They added the King was “shocked and saddened” to learn about the death and would be in touch with the family to share his personal condolences.
Sullivan joined the army aged 18, starting at the Army Training Centre in Pirbright, Surrey, before joining the King’s Troop Royal Horse Artillery in 2021, the Ministry of Defence said in Monday’s update.
Her commanding officer said Sullivan, who was known as Sully to her friends, was a “fearless and gifted horsewoman”.
She took part in ceremonies for the funeral of Queen Elizabeth II in 2022, as well as King Charles’ coronation in 2023 and had recently qualified as an advanced regimental riding instructor.
Lt Gen Mike Elviss, master gunner at St James’s Palace, said Sullivan was a “fine soldier” who “died doing a job she loved surrounded by people who held her in the highest regard”.
“The Royal Regiment of Artillery and the wider British Army is a lesser place without her. I could not be more sorry nor saddened by her loss,” he added.
Defence Secretary John Healey said she was “a brilliant young soldier who served our nation with dedication.
“We’re all deeply shocked and saddened by her death.”
Thames Valley Police said officers “have not found any suspicious circumstances” surrounding the death.
The annual Royal Windsor Horse Show sees international equestrian competitions and military displays, take place on the grounds of Windsor Castle.