While there are spells related to weather like:
* Aguamenti: This charm produces a jet of water, which could be useful for summoning water in a drought, but it's not a rain-making spell.
* Nimbus: This spell summons a cloud of fog or mist, but it's not associated with rain.
The ability to control the weather on such a grand scale is never depicted in the books. However, characters do utilize magic to manipulate weather phenomena like:
* Hermione Granger: Uses a charm to make the tent dry during the Triwizard Tournament.
* The Dementors: Generate freezing fog and chilling air as part of their nature.
It's possible that a spell to start raining exists within the Harry Potter universe, but it's never explicitly mentioned or used.