The power vacuum was filled by the same structures that had caused it in the first place - that is, Roman authority was often replaced by the Germanic peoples/kingdoms that had caused its erosion.

Precisely how this worked depended on where you were - in Italy, the Ostrogothic Kingdom basically replaced the Imperial state, maintaining temples and public buildings, upholding Roman Law (for Romans - Goths followed Gothic law), and even working with the Senate and the Bishop of Rome. Elsewhere, like in Gaul, the structure of society shifted from an urban one to one closer to what we might think of as the 'agrarian' model. The exact amount of continuity of state and social structure is one of the big debates in the historiography of late antiquity, so you're not going to get a definitive answer on that.

A good brief introduction to this is this article by Peter Heather, which also has a bibliography.

The extreme 'continuity' thesis (that is, Roman society did not so much collapse in this period as shift and transform) is argued in Walter Goffart's Barbarians and Romans, AD 418-584: The Techniques of Accommodation. The opposing view, that the fall of Rome was a collapse leaving a power vacuum, is what Bryan Ward-Perkins' The Fall of Rome and the End of Civilization argues. Those texts aren't directly about the subject of your question, but they deal with it in a detailed manner.