Did a quick read about this, seems all you need to do is a HTTP PUT to the path where you want to put the file, the PUT body should be the file, you may also need to include the file type in the Content-Type header.
PUT On Session "data can be a dictionary, list of tuples, bytes, or file-like object. If you want to pass a json body pass a dictionary as json parameter."
Even though this is for curl, this is useful reference for what commands do what in webdav https://www.qed42.com/blog/using-curl-commands-webdav you just need to translate them to requests calls.
Hope that helps,
Dave.