This sort of things cannot be done out of the box. You could hide the menu items in the View menu (and that *may* be enough for a certain class of users) but it won't help for those that memorized the hotkey to call up the panel.
Can't you circumvent the issue by running a script *afterwards* to replace all local formatting with character styles? There are a few of those floating around -- of course it'd need some adjusting to use your own character styles.
To expand a bit on Jongware's suggestion about hiding the menu command to show the character panel you can also customize the Control panel to hide the character controls, and save this as a workspace, then edit the keyboard shortcuts to remove commands you don't want used. I don't recommend this, though. It's easily circumvented by anyone with ID expereince, and a much better approach is to stress to your people how important the use of styles is, and perhaps that their jobs depend on adopting good work habits.
One thing I WOULD recommend is that you add a set of standard character styles to ID as a default. You can make a master document someplace tha has your basic styles, then load those styles with nothing open and they will be available to all new documents, and add them to any templates as well. I think you'll find yo get much better acceptance of using styles if your people don't need to reinvent the wheel for every document they make.