The security challenge only appears on your website for a split second to verify that you are a real user using a real browser to access the page. It won’t block you or anyone in any kind of way if they are using a browser; the security challenge is to verify that you are using a browser, not to verify that you are one single user, so you can let your students use it with no problem.
The only problem is that I’m afraid that if too many students visit your website at the same time and access a lot of pages, it can spike your CPU limit and your daily hit usage, which if you do, it will temporarily suspend your website for 1 day.