Converting a working AJAX form to work with WordPress

You have a typo: wp_ajax_ ajaxfunction. there’s a space after the underscore.

also, use wp_localize_script to set your admin-ajax path:

wp_enqueue_script(
    'contact',
    get_bloginfo('template_directory') . '/subscribe.js',
    array('jquery'),
    '1.0'
);
wp_localize_script(
    'contact',
    'ContactAjax',
    array(
        'ajaxurl' => admin_url( 'admin-ajax.php' )
    )
);

then in your javascript, use ContactAjax.ajaxurl