I have Call of Duty Modern Warefare:3 and have noticed the online vault feature. What does this do? Can I view and download these videos online?
It's in Call of Duty: Elite, the online component of Call of Duty. Login, then go to "Elite TV," then you can "Select a Channel" and then finally, click on your gamertag in that popup. You'll be taken to a place where you can view any video in your vault. (If you're logged into Elite already, this link might take you straight there.)
From this you can also link to YouTube and upload videos from Elite's online vault into YouTube.
You don't need an "Elite subscription" to do this. You can sign up and use these features for free, although there are some other features you have to pay extra for.