Piggybacking off the excommunication question if you need power delegated by a messiah figure to have authority then what exactly is the basis for the Sept institution in Westeros without it?

There’s basically two ways to do it:

• Appointment from above – the most powerful king or warlord chooses his favorite holyperson to be in charge, and they choose the people below them and so forth, similar to the Faith of the Seven post-Jaehaerys and Baelor, or in our world how the Byzantine Emperors appointed the Patriarchs of Constantinople. 
• Election from below – the congregation elects one of their own who’s their favorite holyperson to be their local priest, the priests from a given area elect their superior, and their superiors either belong to a general council and/or elect a single person to lead the whole community. Similar to in our world the conciliarist views of Marsilius of Padua, at least some interpretations of the early Christian church, and the emergence of Presbyterian and similar forms of Protestantism.