It's a little complicated.
There was a River Kingdom before Harren the Black. Rob, while marching with his army, finds his tomb in Storm of Swords, recall. What actually happeend is that the River Kingdom got conquered by Harren from the Iron Islands who then proceeded upon the folly of Harrenhall. So, by the time the Targaryen's show up, no, the River Kingdom is really a non-entity, but it was still a kingdom until shortly before they did arrive with flying lizards with terminal halitosis.
The Riverlands are still treated as a unique socio-political entity to this day, even if it is just about the cruddiest place to live in all of Westeros any time there's a war on. EVER.