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 and primadonna-ish nature. It even liked Brock's homemade Pokémon food. 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. Later they meet the Eevee brothers in The Battling Eevee Brothers who are throwing an evolution party for their younger brother Mikey to evolve his Eevee. They offer Brock a Fire Stone to evolve Vulpix but he refuses.
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. Brock also used Vulpix to help Ash train sometimes (as in EP072, when it fought Pikachu in a training match, resulting in the two of them discovering the ruins of Pokémopolis), and once lent it to Misty when she was competing in the Princess Day Festival and wanted a more balanced team (Ash also let her borrow Pikachu and Bulbasaur for the contest). Vulpix was amongst the Pokémon cloned by Mewtwo in Mewtwo Strikes Back and fought its clone, before Mewtwo took all the clones away with him.
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.
Since Brock mentioned in Beauty and the Breeder that he always knew that Vulpix never really was his Pokémon to begin with, even though he explicitly caught it with a Poké Ball in its debut episode, it can be considered the only Pokémon to not be owned by a main character despite being placed in said trainer's Poké Ball.
For more information on this Pokémon's species, see Vulpix.