What type of video file did you upload? Do you have the multimedia filer on?
Students can always upload to youtube, and paste the URL in the text area. When view by reviewers, the youtube player will be embedded as long as the multimedia filter is on.
I tried it with an mp4 and .mov.
We're trying to make the process as streamlined as possible, and the videos aren't meant for public access. Maybe there is a video plug-in that can be used for Workshop?
Rob is right. If you let students to embed a multimedia file (such as video or audio) into the submission text, the multimedia filer will display an instant player for it. In any case (with or without the filter enabled), users should be able to download the uploaded file.