How Can I Add Code To A Preexisting Function Without Editing it

This would depend on the way the plugin was structured. If it is just procedural then your pretty much out of luck. If it is written within a class you can extend or create a new instance of it and override methods. As an example in my media tools plugin I use a portion of Regenerate Thumbnails by calling

 if ( class_exists( 'Regenerate_Thumbnails' ) {
        $regen = new Regenerate_Thumbnails();

        $regen->class_method();

        $regen->another_method();  
    }
}

To override a method:

 if ( class_exists( 'Regenerate_Thumbnails' ) {

   My_Class extends Regenerate_Thumbnails {

         function __construct() {
            parent::__construct();
         }

     parent_function_to_override() {
        //do stuff
        }
}
}

$barorfoo = new My_Class();
$barorfoo->parent_function_to_override();

Again this totally depends on the way the parent plugin is constructed.