Vulpix was given to Brock by Suzy in the episode Pokémon Fashion Flash. Suzy was Brock's idol, and Vulpix took a liking to him, which was surprising, as the little Pokémon had a very snobbish nature. It even liked Brock's homemade Pokémon food. It drove away Team Rocket with a very powerful Fire Spin, driving home the point that a focus on outer beauty could eclipse a Pokémon's inner strength. Impressed with the way Brock cared for his Geodude and Ash's Pikachu, Suzy decided that Brock might be able to take better care of Vulpix than she could.
Like many of Brock's Pokémon, Vulpix received very little screentime compared to the Pokémon of other main characters. When it was featured, Brock was often seen grooming and pampering it. This let to events in The Ninja Poké-Showdown, where Misty offered to trade her Psyduck for Vulpix, being sick of her Psyduck's headache and annoyed at Brock's compliments of Vulpix. Later the group met the Eevee brothers in The Battling Eevee Brothers who were throwing an evolution party for their younger brother Mikey to evolve his Eevee. They offered Brock a Fire Stone to evolve Vulpix but he refused, saying he didn't want to force Vulpix to evolve if it didn't want to.
Brock also used Vulpix to help Ash train sometimes, as in The Ancient Puzzle of Pokémopolis, when it fought Pikachu in a training match, resulting in the two of them discovering the ruins of Pokémopolis. In this battle, it was revealed that Vulpix could use Quick Attack.
Vulpix was the only of Brock's Pokémon cloned by Mewtwo in Mewtwo Strikes Back and briefly fought its clone, before Mewtwo took all the clones away with him.
Vulpix battled Miki's Skarmory in Hot Matches!. Its Flamethrower was dodged by Skarmory via Agility. Vulpix attempted to outrun Skarmory with its own speed, but Skarmory used Steel Wing on Vulpix, causing massive damage to Vulpix and knocking it out.
Brock later met up with Suzy again in Johto and returned Vulpix to her in Beauty and the Breeder, saying at the time that he had always felt that Vulpix wasn't really his and he was just looking after it for a time.
Brock uses a Vulpix in Round 1 of the original Pokémon Stadium, and a Ninetales in Round 2. This is the only time Brock has possessed a Vulpix in his game incarnations.
Since Brock mentioned in Beauty and the Breeder that he always knew it never really was his Pokémon to begin with, even though he explicitly caught it with a Poké Ball in its debut episode, Vulpix can be considered the only Pokémon to not be owned by a main character, despite being placed in said Trainer's Poké Ball.
In its debut episode, Vulpix was able to have been caught by one of Brock's Poké Balls without any problem, even though it was Suzy's Pokémon.
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For more information on this Pokémon's species, see Vulpix.