If they are internal users then you have a few options. You can utilize SharePoint lists and turn on list item permisisons where users only see their own entry's and owners see all and link them to it.
You could also utilize the new Forms feature where you can have them upload attachments to a form submission and grab them there. This works for internal users only currently so should be fine in your use case. You could even tie a Flow to them to move the attachments to a destination folder and mark a column with who submitted etc.
You could also build a flow to move files over immediately apon upload to a library into a secure location and tag the name of the uploader to them.
Anyway, just a few alliterative idea's that would satisfy the need since we've had these options for awhile now and use them internally fine, external has always been the issue and the new Files upload in OneDrive solves this gap.