A runtime error 438 suggests that it doesn't support this method.

Sub RemoveOrgChartPictures()
    Dim vsoPage As Visio.Page
    Dim vsoShape As Visio.Shape
    Dim intFillType As Integer
    For Each vsoPage In ThisDocument.Pages
        For Each vsoShape In vsoPage.Shapes
            If vsoShape.Master.Name Like "*Org Chart*"Then
                ' Check if the shape has a fill format
                If Not vsoShape.Fill Is Nothing Then
                    ' Get the current fill type
                    intFillType = vsoShape.Fill.Type
                    ' Check if the fill type is compatible with Solid
                    If intFillType = visFillSolid Or intFillType = visFillPattern Then 
                        ' Set the fill to solid
                        On Error Resume Next 
                        ' Ignore potential errors            
                        vsoShape.Fill.Solid ' <---- Trying to bludgeon this with a hammer.
                        On Error GoTo 0 
                    End If
                End If
            End If
        Next vsoShape
    Next vsoPage
End Sub
If this does not work, then sorry, for I am out of ideas.