The VAT number of a client, if valid, is added to customer’s billing address. Thanks to that, it’s displayed automatically on the My Account page, as well as on the PDF invoices. Here’s an example of an invoice generated by the PDF Invoices & Packing Slips plugin you mentioned: https://snipboard.io/gzeMHv.jpg
If that value doesn’t appear, perhaps there’s another plugin, or some code, altering the billing address, removing the VAT number. Going by exclusion, disabling one plugin at a time (except WooCommerce and the Aelia ones) can help identifying the possible source of the conflict.
Technical information
The EU VAT Assistant uses a filter for hook woocommerce_order_formatted_billing_address
to add the VAT number to the billing address. We would suggest to check your plugins, as well as the theme, to see if there might some code running a similar filter and removing the VAT number, either deliberately, or accidentally. If that’s indeed the case, its authors should be able to tweak the code so that they preserve the VAT number field, which will then appear on the invoice.