I think LoC has partnered with the Internet Archive in the past, but now they're planning a much more ambitious version at http://www.netpreserve.org/about/index.php

LoC is actually relatively forward-thinking when it comes to digital archiving. For example, for a while now they've been encouraging people to digitize archives of their own personal data: http://www.digitalpreservation.gov/personalarchiving/

You could also argue that as a government agency the LoC should prioritize important archiving tasks that aren’t already being done by the private/non-profit sector. Despite being a corporate controlled medium, Twitter is still an important communication medium and should be archived. For whatever reason, others aren’t doing it, so it’s an obvious thing for LoC to do.