The single word "Sig1" in your column is not the bookmark but the bookmark range. You don't need to delete the bookmark. Just redefine its range and recreate it.
Code:
Dim oILS As InlineShape
With ActiveDocument
If .Bookmarks.Exists("Sig1") = True Then
With .Bookmarks("Sig1")
Set oILS = .Range.InlineShapes.AddPicture(FileName:="C:\Sig1.png")
.Range.Bookmarks.Add "Sig1", oILS.Range
End With
ElseIf .Bookmarks.Exists("Sig2") = True Then
With .Bookmarks("Sig2")
Set oILS = .Range.InlineShapes.AddPicture(FileName:="C:\Sig2.png")
.Range.Bookmarks.Add "Sig2", oILS.Range
End With
End If
End With