Hi,
I don't see any problem in your message.
If you want an "invoice me" payment method during the checkout, just create an offiline payment method with the "check" or "collect on delivery" payment methods. The customer would then select it during the checkout and when finishing the checkout, the order would be created with an email notification sent both to him and the vendor. Then the vendor could simply modify the order to create the invoice and have the business pay it offline.
In such case, you might want to customize the emails text with translation overrides:
hikashop.com/download/languages.html#modify
And you can go even further and customize the emails based on the payment method selected by editing the emails via the menu System>Emails (Business edition only) and using such code:
<?php if($data->order_payment_method=='collectondelivery'){ ?>
my custom text for orders made with the "invoice me" payment method which uses the collectondelivery payment plugin
<?php } ?>