Yes, the OneDrive app has an automatic photo backup option.
That being said, one quirk I've noticed with the app is that if you save pictures from messages or a shared photo stream that inserts the saved photos before whatever your latest camera photo roll is (date-wise), OneDrive will not recognize the new photos added and won't automatically upload them. You can still easily upload the photos through the OneDrive app, but you have to do it yourself vs. automatically. It seems that OneDrive just looks for photos newer than what it tags as your previous newest photo before it uploads it.