How can I generate an invoice using the purchaseid?

This feature is not part of the shop styling plugin.
But you I can give you a few hints:

The files are located in a directory that is not accessible because of a htaccess file.
You could create a PHP file in your theme directory to read an invoice from this directory and send it to the browser.

<?php
header('Content-disposition: attachment; filename=invoice.pdf);
header('Content-type: application/pdf');
readfile(HAET_INVOICE_PATH.invoice.pdf);
?>

This file could use the purchaseid as a parameter to get the filename.

BUT you need to check the owner of the invoice before sending the output to the user.

But how can I generate that invoice file?

You can’t generate invoices for orders sent before the shop styling plugin was installed, because the invoice needs data from the shopping cart.

I am late for the answer, sorry.

Yes, I know that, the order is sent and the wp_wpsc_purchase_logs table has got a row for the order.

Now I want to generate the invoice programatically, using a php function I suppose, giving the purchase_id or similar value.

I want to get a button in the private user area to generate the invoice giving the purchase log id.

I understand what you want to do but unfortunately the shop styling plugin relies on some computations, that are only available in the cart and will be deleted after checkout.
I think your idea is possible, but there is currently no plugin available to do this. If you need help I can have a look at your store and give you a quote for the changes. My email address is here: https://etzelstorfer.com/kontakt/

Do you think that the cart contents is deleted after checkout?

In the dashboard when you go to (wp-admin/index.php?page=wpsc-purchase-logs), you see the sales, and when you click on one sale, then, inside, you can always see packing slip.

You can do this:

wp-admin/index.php?page=wpsc-purchase-logs&c=packing_slip&id=[PURCHASE_ID]

But the output format is html, not pdf.