This is an interesting conundrum on something I haven't ever explored in Word.
I found a clue on
ExcelForum by a user called marvinparanoidandroid which greatly helps. Unfortunately you will likely be required to create a free account there to see his answer.
Can you try this code on your document?
Code:
With ActiveDocument.Signatures
.Subset = msoSignatureSubsetSignatureLinesUnsigned
If .Count > 0 Then 'at least one unsigned signature in doc
.Item(1).Sign
End If
End With