I would assume that you're looking for this to make file transfers to/from your Wii easier? Or perhaps for keeping multiple homebrewed Wii's synchronized? I don't think there's anything that specifically incorporates cloud storage, however, I do use FTPii for transferring files to/from my Wii. A little buggy, but it gets the job done.
You could also look at WiiXplorer, which, according to the docs, has a FTP server AND client built in. I haven't used it, but, theoretically, this should allow you to transfer files between Wii A and Wii B, but not to Wii U! ;-) Hahaha, I couldn't resist.
If I'm way off base here with my assumed motivations, then please let me know.