anyone know how to keep iPads from allowing students to open up new internet windows when taking quizzes on ipads?
I am guessing the same way you do on a desktop. I found that respond us has an iPad app for their lockdown browser. https://support.atu.edu/link/portal/16084/16101/Article/819/Can-an-iPad-be-used-with-LockDown-Browser references it's use at one institution.
I don't think there is a way a website can restrict this more than the restriction settings on a quiz. see http://docs.moodle.org/26/en/Quiz_settings#Display
Keep in mind that you can shoot the iPad with a machine gun and that wouldn't prevent the the user from opening a new window on the phone that's in their pocket, or their smartwatch. You can do some things like Jason mentioned, but the result is only that you make it slightly less convenient to Google the answers. You'll never get close to making it impossible.