Order by value of Custom Field using url string

Try use get_query_var instead of $_GET and also increase priority

function my_pre_get_posts( $wp_query ) {

// do not modify queries in the admin
if( is_admin() ) {
    return $wp_query;
}

// allow the url to alter the query
if( isset(get_query_var( 'budget' )) ) {  
    $wp_query->set('meta_key', 'budget');
    $wp_query->set('meta_value', get_query_var( 'budget' ));  
} 

// return
return $wp_query;
}

add_action('pre_get_posts', 'my_pre_get_posts', 11);