Looks like a content control rather than a legacy field, so you would need
Activedocument.Contentcontrols(i).Range.Text
Where i is the index no. of the control (the first one in the document is 1, etc.)
So either you need to know the number, or you need to use the title or tag of the control (or something else) to identify the control. In the case of content controls, you then have to iterate the collection to find it, e.g.
Dim cc As ContentControl
For Each cc in ActiveDocument.ContentControls
If cc.Tag = "mytag" Then
strDate = cc.Range.Text
Exit For
End If
Next
Be careful how you title/tag your controls. Word does not enforce uniqueness of either titles or tags.
Another way to do this would be to create a Custom XML Part with a "Date" element and connect your control to it using xPath. Then you can retrieve the value directly from the Custom XML Part and you don't need to know anything about the control itself. But I think in a simple form that adds complications that you can do without.