You can use the below code to create dump using "exp" and c#.
The same may work with "expdp" :
protected void btnExport_Click(object sender, EventArgs e)
{
string batchfile = "Docs/batchfile.bat";
string exportfile = Server.MapPath("Docs/myexport.dmp");
string exp_cmd = "exp user/pwd@db file='" + exportfile + "' log=a.log tables=(mytable)";
StreamWriter sw = File.CreateText(Server.MapPath(batchfile));
sw.WriteLine("@echo off");
sw.WriteLine(exp_cmd);
sw.WriteLine("exit");
sw.Close();
System.Diagnostics.Process p = System.Diagnostics.Process.Start(Server.MapPath(batchfile));
p.WaitForExit();
p.Close();
Label2.Text = "Exported to : "+exportfile;
}