Hi Roy,
Which version of Turnitin are you using? I know that we have the Direct Integration with Moodle 2.2. We have some tutors who ask for work to be submitted via the Moodle assignment, and then save the student submissions onto their PC or wherever before marking. If they suspect plagiarism it's possible to set up a Turnitin assignment on the same course (somewhere all the students are enrolled), and then submit the paper 'on behalf of' a student (you could always hide the Turnitin submission). You choose the student's name from the list of enrolled students.
Alternatively, if you want all the students to see the originality report, you could always do this for all of the already submitted papers, or simply ask them to resubmit to the Turnitin assignment.
Thanks,
Alicia