Fire curl command everytime new user is created

//This action hook allows you to access data for a new user immediately after they are added to the database. The user id is passed to hook as an argument.
    do_action( 'user_register', $user_id );

so you could:

add_action('user_register','my_user_register_function');

function my_user_register_function($user_id)
{
$user = get_user_by( 'ID', $user_id );
//your CURL stuff
}