One simple way to achieve that is create an application pool for the web app with a service account, and grant the permission for that service account inside the SQL server, then wrap all ADO.NET code inside a System.Web.Hosting.HostingEnvironment.Impersonate() block:
using (System.Web.Hosting.HostingEnvironment.Impersonate())
{
using (SqlConnection connection = new SqlConnection(connString))
{
SqlCommand cmd = new SqlCommand(spStoredProcedureName, connection);
//... other ADO.NET code
}
}