@lchallen ,
Getting a link like this to actually do anything would require adding JavaScript to the page, which would have to be done through the custom global JavaScript in the Theme Editor.
You can make read-only API calls directly from Canvas. For example, if I go into a content page and add a link to /api/v1/courses/1234/users, then clicking that link authenticates as the user viewing the page and shows the list of users (if they can see it) for the course.
That said, it's also going to come with a while(1); in front of it and just return JSON code. It will not give you a downloadable ZIP file.
The content export routine you're looking at is a POST, not a GET, so you wouldn't be able to call it directly from within a page. Any HTML necessary to do that would get blocked by the Canvas HTML Editor Whitelist.
Also, it takes some time to prepare the list of files as some may be very large. It might not even start right away based on how busy the system is doing other things. The documentation says that you would need to monitor the progress of the job. Those kinds of things are not suitable for a situation where someone is sitting, waiting on the page -- especially a student. It may be okay in most cases, but it's something to consider.
All of the stuff mentioned that would be involved in that would make it unsuitable for generating a link or button within the page that zipped it while the student waited.
You might be able to write an LTI that would do it. It could scan for the latest versions of the files and zip them up. You could, possibly, provide the link for just the instructor that would trigger an update of a zip file in an offline process. Again, downloading all the files, zipping them, re-uploading that is not something that can reliably occur in real-time.
If this is just for a single course, then it really is going to be easiest for the instructor to mirror the Canvas structure on a local computer and update the zip files when the instructor changes one of the files. This could be automated, even to the point of using a cURL to upload the updated zip file to Canvas. WebDAV would make that easier and it has been suggested a couple of times, someone even asked about an integration with Dreamweaver once, but all of the feature requests didn't gain enough momentum and are currently in cold storage.
You only need to link to the zip file once, though. When you re-upload the zip file containing all of the files, then you replace the existing once and the link is still good.
Any solution that you try to come up with automating the process within Canvas is going to be a lot of work. For just one or two courses, you have to weigh the return on investment and it's easier to just maintain the zip files separately. If you don't need access to all of the individual files, it would be easier for the students to just download a zip rather. That would also be less for the instructor to upload. Of course, if these multiple files contain content needs to be displayed within Canvas, then keeping just the zip file isn't an option.