Items can only be placed in the Safari Zone after beating the Elite Four. After beating them, you automatically have access to the items to place where you want upon entering the Safari Zone.
Azurill can also be obtained by breeding Marill or Azumarill while holding a Sea Incense, which can only be obtained post-game (in Cerulean Cave).
The last way to get Azurill in HGSS is through the Pokewalker, on a route that, once again, is only available post-game.
So, no, there is no way to get Azurill before beating the Elite Four in HGSS. (without trading)