Yes, you can! Submissions can be downloaded via the Submissions tab in the Game Manger.
Near the top of the Submissions page, there is a "Download Submissions" button (as shown below). Clicking this button will launch the Submissions Download wizard, which can be used to download all game submissions to your computer as a .zip file. This .zip file will include all photo and videos submissions, as well as a spreadsheet for text and GPS missions. Optionally, submissions can be grouped by mission or by team.
Note: The "Download Submissions" button will only be visible once the game has received submissions from at least one participant.