That's right with iOS 6 you have "the ability to use Upload , Select File , or Choose File buttons and links found on websites for submitting files."
Moving files has not been easy when there are multiple users of an iPad in school environments. Email often presents issues due to size limts (and the same would hold true with video).
This post provides details about how to do this, and also looks at the process to add media to an Edmodo post. There are also suggestions on ways to collect student work.
Via Beth Dichter, Aki Puustinen