I have created a layout in Pro that includes dynamic text for the author which should display the Portal user login name:
<dyn type="layout" property="metadata" attribute="contactname" emptyStr=""/>
I published the layout as a web tool. However, from Web AppBuilder 10.6, the PDF that is created lists "Web AppBuilder for ArcGIS" as the author name instead of the user login name.
WAB print widget passes the value from 'Author' text box to the print service. This value gets used in the dynamic text element on the layout that you mentioned in your email.
Print service does not have any knowledge of who is the user is on the client side.
The user can add their name in Author text box or you might be able to add some custom logic in WAB print widget to auto populate this text box with name of the user currently using this app.