@cmulligan ...
Canvas does not have any built-in ways to disable the Canvas Inbox (which is not the same as e-mail) or other outside applications such as Outlook to stop students from contacting each other while taking a quiz/test/exam in Canvas. You might consider looking into the Respondus Lockdown Browser as an option...which is a third-party integration with Canvas which prevents students from going to other screens while taking a quiz/test/exam. However, this may not stop students from using their mobile devices to message their classmates...so that's something to keep in mind as well.
Sorry I don't have better news for you, but I hope this helps to answer your question.