The Stormlands was, in terms of land and vassal houses, probably the smallest of the Seven Kingdoms. The Iron Islands are smaller, but they're technically not one of the 7.
Plus, it was the Kingdom probably still most loyal to the Targaryens, as Dragonstone used to be their seat. Robert may have been seen as a Usurper there as well. In the books, after he takes the throne, Stannis has to then go separately to quell the Stormlands and take Dragonstone.