Doing this would entail writing an application event procedure, for which see:
https://wordmvp.com/FAQs/MacrosVBA/AppClassEvents.htm
and adding that to your document's template.
Once you've done the basic setup, you'd insert the following code into the class module:
Code:
Private Sub oApp_WindowSelectionChange(ByVal Sel As Selection)
If Sel.Information(wdWithInTable) <> True Then Exit Sub
If Sel.Tables(1).Range <> ActiveDocument.Tables(1).Range Then Exit Sub
Dim r As Long, c As Long
Application.ScreenUpdating = False
With Sel
r = .Cells(1).RowIndex
With .Tables(1)
If Len(.Cell(r, 1).Range.Text) > 2 Then Exit Sub
If r > 1 Then
If Len(.Cell(r - 1, 1).Range.Text) = 2 Then Exit Sub
If Len(.Cell(r - 1, 2).Range.Text) = 2 Then Exit Sub
End If
.Cell(r, 1).Range.Text = Format(Now(), "hh:mm:ss")
End With
End With
Application.ScreenUpdating = True
End Sub
As coded, the macro will only insert the timestamp into the first column of the first table in the document. After the first row, it triggers only if the first two cells of the preceding row have content.