WiiConnect24—the standby service that fetched messages, Miis, and updates while your Wii glowed orange—was permanently shut down on June 20, 2013. It wasn't a downloadable app; it was baked into the Wii OS. When Nintendo killed Wii online services, WiiConnect24 died with them.
Original WiiConnect24 is gone forever. RiiConnect24 is the closest you'll get—and it's impressive for a fan project.