You should be able to allow students to record right in the quiz. First, as you are creating your quiz question you'll want to enable to Rich Content Editor in the Options dropdown.

Then when the student is taking the quiz, they'll be able to click the multimedia link and select Upload Image.

This will display a popup window where they can 1. click on the Record tab (instead of the Computer tab), 2. Click on Webcam and select No Video - unless you would like them to record a video of them pronouncing the work, which might actually be a pretty good approach, 3. If they have multiple microphones then they'll want to select the one they would like to use.

With all that taken care of, they can click the big blue Start Recording button.