No. Moderators cannot change themes.
It would be possible to create a theme that would, say, make the banner a photo in a topic with a certain tag or category (or the newest topic with that tag or category). You could then limit access to that tag or category to members of a certain group. I wrote a plugin once that took its data from the most recent reply to a topic.
So, without looking at the code, I think you could fork the theme and do something like that.