Call function without having to wait on response

When submitting a form I want to call an additional function that will connect to a webservice. Because of the slow response time of that webservice, I don’t want the page to keep loading untill the process is done.

Instead the action should be fired/called and run on the background.

What is the best way to achieve this?

Use the WordPress HTTP API with a low timeout & blocking disabled. It’s what WordPress core does to spawn the cron API:

wp_remote_get(
    $url,
    array(
        'timeout'   => 0.01,
        'blocking'  => false,
        'sslverify' => false,
    )
);