I've been doing some research on this and I'm still not sure if it's possible.
Is there any way to retrieve a user's bookmarks and display them on the page without violating their privacy? I saw these two threads talking about using JS to do this:
How to get Bookmarks toolbar information in JavaScript code?
Show all bookmarks using javascript
I'm not thinking about doing this automatically, users would have to opt-in with a button that said "Display Bookmarks from My Browser" or something similar.
Is it possible?
Not without browser extensions. You could write a browser extension for the major browsers which retrieves the information and feeds it back to your page.