I'm not sure if this is explicit in the original Frank L Baum The Wonderful Wizard of Oz or the 1939 film The Wizard of Oz.
However in the book she does carry an umbrella, instead of a broom as in the film, a very convenient thing for someone allergic to water, and possibly more than a coincidence, almost as though she didn't want to get wet...
Once the Witch struck Toto a blow with her umbrella and the brave little dog flew at her and bit her leg in return.
- The Wonderful Wizard of Oz - Chapter 12 - The Search for the Wicked Witch
The wicked witch of the south Singra in the book The Lost King of Oz is afraid of water killing her, though whether this is because she know it's bad for them or because of what happened to the wicked witch of the West and Mombi is open for speculation.
As the canonicity of Oz is pretty fluid due to so many people putting their spin on things I'm also going to mention the 2013 film Oz The Great and Powerful Theodora who becomes the wicked witch knows water is dangerous to her as she is burnt by her own tears, even before becoming wicked.
And a further twist from further along the canonicity scale, in the musical Wicked which I have recently seen for the first time:
The weakness to water is something that has been made up and Elphaba (the name of the Wicked Witch in this iteration) uses the belief everyone has in this to fake her own death to be with the man she loves.
