The application will have to build the Export Map request and send it to the REST endpoint. Here is the reference for that type of request.
Export Map (Operation):
http://resources.esri.com/help/9.3/arcgisserver/apis/rest/index.html?export.html
here you can see how call rest from .net
http://rexdotnet.blogspot.com/2009/11/using-arcgis-server-rest-api-in-net.html