The 403 code will appear for a site that recognizes authentication, but refuses to grant access for reasons known only to it. Make certain that you are not using any potentially conflicting Safari Extensions, Internet plug-ins and other add-ons (all addressed in the Support document), and also confirm that you have no active proxy configurations (Network Preferences > Advanced > Proxies).
If none of that resolves the problem the next logical step is to launch Keychain Access and filter for appleid. Delete anything conforming to the Apple ID you are using when you attempt to connect to those sites. You will need to provide your Apple ID credentials again, but you only have to do that once.
In the end there may be nothing wrong with your Mac. Since leroydouglas experienced a similar problem Apple may just be overwhelmed at the moment, and perhaps the reason is related to today's Sierra update. I haven't had any trouble trying to connect to any support site on any of my Macs, but one data point means nothing.