it does seem strange that admin users work but students don't - perhaps you have some custom roles or modified permissions on your site that cause this.
there is a a capability within scorm called "savetracks" - make sure the role that your students are in have this capability (it is added by default to the student role in Moodle)
but - I would only expect that to be the case if no user data is saved when students are accessing the SCORM - if some data is saved but not essay/short answer questions that suggests a different issue. If you can reproduce this you could turn the SCORM debugger on, and then post the details from the debug log here for us to look at.