Creating loop within functions.php

Just loop over WP_Query::$posts to get the titles:

$option_posts = new WP_Query( [ 'post_type' => 'organisationer' ] );
$options      = [];
foreach ( $option_posts->posts as $post ) {

    $key = "option_{$post->ID}";
    $options[ $key ] = apply_filters( 'the_title', $post->post_title );
}

woocommerce_form_field_radio( 
    'custom_field', 
    array(
        /* … */
        'options' => $options
    ),
    $checkout->get_value( 'custom_field' )
);

I’m not sure what exactly woocommerce_form_field_radio expects as parameter. Maybe you have to flip the key and value in the associative array $option. But this is basically how to get a list of post titles from a query object. You may also consider to remove the the_title filter, depending on your needs.