Turn off Focused Inbox for my company without using PowerShell

Why can't you use Powershell? This is the approach recommended by MS. It is run once against your organization to disable focused inbox for everyone. Any admin with the necessary permissions can run it.

The only alternative would be to use GPO to configure it but Outlook doesn't appear to include a policy for the focused inbox, because it is set at the organization level. Even if they did then it would only impact the desktop client and not outlook online I'd wager.

Behind the scenes, at least with the current Outlook client (might not work for the new one) then it is backed by the registry. You can create a GPO to update user's registry at login. This isn't really recommended given that it slows things down and PS would be better but if you wanted to go that route then the key appears to be: HKCU\Software\Microsoft\Office\Outlook\Settings\Data. Within this key is a value called <email_address>_IsFocusedInboxEnabled. You'll have to dynamically set the value to the user's email address since it is per account. The value itself is a JSON value. You can copy an existing value as an example. Within this JSON is the actual setting value in the form: "value": true. Set it to false to disable the focused inbox.