Does the story, art, and sound team have any part in deciding which skins are worked on?

Well there is no specific "Story, art and sound" team. There's a narrative team which focuses on the writing side of things; lore, stories, world building, etc. Art has a bunch of different areas, and some of that involves skins. Sound is involved in a lot of things too, some of it being skin related.

Basically Story, art and sound is a catch all for anything of a narrative, artistic or music nature for fans to discuss. There's no specific team for all of them, but individual teams for the bits and bobs. And the folks on this board don't really have any say in what skins come when, as that's Riot's prerogative, but feedback is often provided here.

I wouldn't say the lore department could actually dictate to the skins team to make a new skin on the fly.

That being said, however, I'm sure that the skins team pays close attention to what foundations is working in order to find inspiration for new skins, and would try to coordinate it's release with any lore events they have planed.

(Captain Miss fortune and Gangplank for example)