Hi Michael - the "Turnitin Dan Marsden Moodle integration" (don't ask me who called it that...) integrates with existing Moodle plugins such as the Moodle assignment module, workshop and forum modules. It adds an extra setting when creating the activity in your course that allows you to enable Turnitin within that activity.
The standard Moodle Assignment module (since 2.3) contains the feature you are looking for - and you can get the student submissions to send to Turnitin by using the "Dan Marsden Moodle integration" - the code for that plugin is available here:
https://moodle.org/plugins/view.php?plugin=plagiarism_turnitin
BUT - if you aren't using that plugin already (which I presume you aren't) - you shouldn't install it as there is a new plugin coming from Turnitin that provides the same functionality but is released and supported directly by Turnitin - there are still a few small rough edges which should be ironed out within the next few weeks and there isn't any documentation for the new plugin yet (AFAIK) but that should come soon too.
More info on the new code is available via my blog post here:
http://danmarsden.com/blog/2014/01/15/new-turnitin-api/