I was wrong, it looks like that field isn't available. I looked at the API explorer and fusion and there is no entry date gathered for time off. We handle time off through a project and tasks for our team and create a custom calendar.
We had that problem originally. Check to see if you have a drop down menu at the top left corner of the support screen. We have to change profiles from that drop down to log a ticket since our Admins have 2 different Adobe profiles (one for Workfront and another for Creative Cloud).
If you have Fusion, you could have a field on the request where they enter the information and then create a scenario where Fusion transfers that information to the project fields.
Does the user have manage access to the report they're trying to send or just view access? They also need edit access to
Edit access to Reports, Dashboards, Calendars and Edit access to Filters, Views, Groupings.