Moodle can be set up for assignments that only require one member of a group to submit a document.  Grading the assignment will send the same grade to all members of the group.  

Configuration

Set up groups

• Go to Participants (left menu) → Gear icon (top right) → Groups
• Click on "Create Group"
• Give the group a name (e.g. "Group 1" or "Monday Lab A") and save
• Select the newly created group and click on Add/remove users.  Add users as needed to this group.
• Repeat the process for all the groups in the course

Set up the assignment

Create a Moodle assignment as usual, but adjust the settings for group submission.

Under "Group submission settings" section

• Students submit in groups → Yes

Under "Common Module settings" section

• Group mode: Visible groups

How it works

Students will submit files to the assignment as usual, but when one group member submits a file, it will show up under all the group members' submissions.  Grading one group member will send the grade to all group members.