The world No.1 scored the lowest-ever points for the three-day event's opening discipline.
Jonelle Price and husband Tim have both claimed prestigious wins in 2022.
The daughter of 1998 Olympic medallist Tinks Pottinger is following in mum's footsteps.
Sir Mark Todd's four-month suspension is already behind him.
The eight-time Olympian has already served two months, while the final two months of his ban are suspended.
The British Horseracing Authority suspended the two-time gold medallist last month.
The rider was given a yellow card and disqualified after one event.
Sir Mark Todd competed at eight Olympics, winning two gold medals in three-day eventing.
The incident posted to TikTok falls outside the time-sensitive window needed to charge the New Zealand Olympian.
The animal welfare group are also again calling for an end to equestrian events at the Olympics.
The British Horseracing Authority says the footage has "rightly caused anger and upset within the equestrian community and beyond".
But animal rights campaigners wonder how often the Kiwi icon has committed such acts over his career.
The clip posted to TikTok shows Sir Mark hitting a horse as it stops in front of water.
Brougham competed in dressage at the 2016 Olympics and 2018 World Equestrian Games.
The animal welfare groups cite cruelty to horses as a reason to move on from equestrian competition.
Tim Price and Vitali collected three rails, dropping from fifth to 16th in individual standings.
With the showjumping to come, the NZ riders lurk just outside the medals at Tokyo.
The NZ team will contest athletics, equestrian, golf, sailing and weightlifting on Sunday.
Aussie equestrian rider Jamie Kermond has been provisionally suspended in Tokyo.
Tim and Jonelle Price spearhead the Kiwi eventing team for Tokyo.