I think the closest we've gotten to that sort of story so far is the first season of Rebels. While it is about a group of people fighting against the Empire, the entire season is basically contained to the planet Lothal. As the main characters hit the Imperials where and how they can, you get small glimpses of how the regime is affecting the little guy - a disgruntled shopkeeper cowering at the barrels of stormtroopers' blasters, a farmer and his family taken prisoner when the refused to give up their farm, a kid joining the Imperial Academy because his sister did the same and was never seen again. That was really one of the strengths of the season for me, so I'm not really sure how I feel about the series expanding toward the scale of the Galactic Civil War next season.