C1WebChart renders as an image on the webpage, so you could just right-click and select Save image as...
If you want to capture that save-as functionality in a button click event, see my attached sample.
If you want to save the chart into a PDF, you'll have to use C1PrintDocument or C1PdfDocument to create the pdf. You can add the chart as an image from GetImage() to the pdf document before saving or viewing. This is also shown in my attached sample.
Hope it helps,
Greg L