No, the Flex API doesn't support reading shapefiles.
In addition, any browser application will generally not be able to read files directly from your local file system (for security reasons).
However, I think some people have extended the API with custom code to read shape files, but the shape files still need to be on a web server (or uploaded). Alternatively, you can (with the Flex API) use an Adobe AIR application (which works more like a desktop application) to be able to read local files. However, you would still need some custom code to read the shapefiles.
