So it's the chart class, not the chartobject class which has an export method. This worked fine for me (I set up a chart before running the code):
Code:
Sub foo()
Dim cto As ChartObject
Dim cht As Chart
'get a reference to the chartobject on the sheet
Set cto = Sheet1.ChartObjects(1)
'get a reference to the chart
Set cht = cto.Chart
'export the chart as a jpg
cht.Export "C:\Temp\MyChartPic.jpg", "JPG"
End Sub