Does JotForm allow for an outside user to create a log-in and save responses?

For example, if a student is applying to a university and they are adding content such as essays, and attachments such as transcripts. Can the form be set up by the organization (us) and the user set up a log-in to update their information and submit when complete?

No, login is not possible for each form submitter. If you want each submitter to be able to edit their own submission, or have them saved the filled data before submitting and continue later, then you can apply the following workaround: 

https://www.jotform.com/help/97-How-to-Save-Forms-and-Continue-Later? 

Let us know if you need more help.