And in the aftermath of what Vivec did, how would Azura's servants be able to get to Tamriel if her realm was inaccessable?
That's easy. They came through Mehrunes Dagon's realm, the Deadlands. Moonshadow is closed to Nirn, but that doesn't mean its closed to the other oblivion realms. Indeed, Sul managed to arrive in Azura's realm via another Oblivion realm. I'm sure she was glad for a visitor.
EDIT:
I kinda had a thought that Vivec might have zero-summed after being captured by Mehrunes Dagon's servants during the Oblivion crisis. The dremora may have escorted him from the palace to the Deadlands, where he would await Azura's servants to be escorted to Moonshadow and killed/tortured by Azura. As a last jest, right before being handed over to Azura, he might have released his greedy hold on his individual identity to mock and frustrate Azura one last time. Laughing as he faded into nothing.
No real evidence for this of course. Just how I imagine Vivec's going. Though who knows? Maybe he decided to travel with the Nerevarine to Akavir? Or maybe he just zero-summed unintentionally while meditating in his palace?