I have a web application that\'s branded according to the user that\'s currently logged in. I\'d like to change the favicon of the page to be the logo of the private label,
Yes totally possible
e.g.
Then whatever server side language / framework you use should easily be able to find the file based on the userId and serve it up in response to that request.
But to do favicons properly (its actually a really complex subject) please see the answer here https://stackoverflow.com/a/45301651/661584
A lot lot easier than working out all the details yourself.
Enjoy.