Azurill has arrived in Pokémon Sword and Shield, and you can find them in The Isle of Armor expansion pack. You can capture them in a specific location on the island, and you won't have to worry about weather conditions while trying to get it.
You will need to go to Brawler's Cave on the Isle of Armor. You can access it from the Soothing Wetlands, the Challenge Road, or the Training Lowlands. Once inside the cave, you should find one bouncing around the first floor of the cavern. They are relatively common throughout the cave.
A great way to go all the way up and down Brawler Cave is to start at the training lowlands or the Challenge Road entrance and follow it all the way. Following this strict path gives you a direct view of all the Pokemon roaming the cave, increasing the chances of you locating Azurill during your travels.
Azurill is a Normal and Fairy type Pokémon, able to evolve into Marill, then its final evolution is Azurmill. If you want to evolve it, you need to increase your friendship with it, which you can do by having it in your party during Pokémon battles or by feeding it curry at camp.