Can we hook the Customizer on a post or page on the front of the site?

This question is a bit too broad for a Q&A model, but I tend to think that everything you need is already built in. Except that you will want your clients to login, something you can have them do on the front end.

You could even make a custom role which will allow you to hide the whole admin bar from them except for the button that lets them open the theme customizer.

In the theme customizer you can change the capability on settings, which will make some of them available for users with low clearance. In this way you can prevent them accessing settings they have no business with.

You can use the active callback on sections to load different sections for different types of pages (front, archive, singular, and so on).

If you don’t want them to save their actions, hide the save button.