Little is known of the Bazz Brigade history, though what is known is that it was made up of a group of Zora children lead by Bazz, the son of Demon Sergeant Seggin of the Zora Royal Guard Knights. At some point Hylian Champion and Hyrulean Knight Link became a member of the group and would often spend time playing with them and instructed Bazz in the ways of the sword.
According to Mipha's Diary, Link first visited Zora's Domain as a four year old child and was already a prodigal swordsman capable of besting grown adults. It is stated that he befriended Mipha during this period, indicating he presumably joined the group around this time. Bazz presumably started learning swordsmanship from Link around this period due to Link's skill, which was extraordinary for a boy his age. The fact both their fathers were skilled knights, as Link's Father was an accomplished Royal Knight according to Princess Zelda, possibly further helped Link and Bazz to bond due to their similar backgrounds. Young Link was known to be brave to the point of being reckless which, along with his skill and friendship with the Zora Princess, likely endeared him with the local Zora children, including the Bazz Brigade. Link would return to Zora's Domain several times over the course of his life. By the time of his second visit, Link had grown older and was an accomplished Knight of Hyrule capable of wielding the Master Sword.
100 years after the Great Calamity, the Bazz Brigade is shown to have mostly disbanded, with most of its Zora members having grown up with all of them becoming Zora Knights while Link had been sleeping within the Shrine of Resurrection, recovering from the injuries he sustained while defending Princess Zelda from corrupted Guardians during the battle of Fort Hateno during the Great Calamity. Bazz himself, through his training under Link, eventually rose up the ranks to become Captain of the Royal Guard in Zora's Domain. Rivan fathered a daughter whom grew up to be a Zora Knight much like her father, though was apparently born after the Great Calamity as she notes she is less than a hundred years old. Gaddison is the only female member of the group whom was nicknamed The Heroine and is a dedicated knight.
Unlike most of the Zora Elders who blame Link for Zora Champion Mipha's death, members of the Bazz Brigade hold no ill will towards their old friend Link and welcome his aid in stopping the Divine Beast Vah Ruta. They all provide Link with information about what has been going on in Zora's Domain. Bazz and Rivan warn Link to be wary of their respective fathers Seggin and Trello. During the "From the Ground Up" Side Quest, Link can ask Gaddison to come to Tarrey Town, but she declines saying she's a "blood-stained soldier, not a priest" and suggests Link talk to Kapson. Presumably after Calamity Ganon's defeat, Link may have sought their assistance in helping him in rebuilding the Knights of Hyrule as the order had been all but decimated during the Great Calamity with Link the only confirmed survivor. It is strongly implied that Link and Zelda's visit to Zora's Domain after Dark Beast Ganon's defeat is to seek the assistance of the Zora in rebuilding of Hyrule which Link's old friends from the Bazz Brigade would likely support.