Many of the Pokémon you encounter in Pokémon Brilliant Diamond and Brilliant Pearl will have evolution requirements. This means that you have to give them an item, a move, or evolve them while holding something in their hand at a specific time of day. If you're looking to evolve Vulpix into Ninetales, you need to make sure you meet certain criteria.
Before chasing Ninetales, make sure you've caught a Vulpix. It appears in the Great Dungeon in Canyon Caves, Lava Caves, Sand Caves, or Tylpho Cavern when you have unlocked your National PokéDex after completing the Sinnoh PokéDex. It is also important to note that Vulpix is exclusive to Pokémon Shining Pearl. You won't be able to find it in the wild in Pokémon Brilliant Diamond, so you'll need to trade it with a player who owns Shining Pearl.
Now that you have Vulpix, the next step is to give him a Fire Stone. You can find a flint inside the Fuego Ironworks, or you'll need to see it as you explore the Grand Underground. They spawn in the Fossil Digging minigame, so you'll need to complete this minigame to find them. To increase the chances of these items appearing, be sure to collect the Diglett and Dugtrio bonuses you will get by walking through the passages.
Once you have the item, all you need to do is give Vulpix the Firestone and it will immediately evolve into Ninetales.