Upon event I want to execute the creation of an entire website using a website template.
This would be an action to be executed by Sharepoint.
Email with certain subject comes in --> sharepoint creates new website using a template and gives permission to enter the website to the sender of the email
Looks like this is exactly what you need:
Create site by custom template and grant permissions
This article explains how to use Plumsail SharePoint connector, which is a part of Plumsail Actions (third party set of connectors for Microsoft Flow with a fair price).
Thx, much appreciated.