May I suggest to re-read your customer's e-mail?
"(...) in order to be able to pay you, I need invoices issued by a registered company or sole trader showing the address of the company, the tax number of the company, and with the invoice signed and stamped.
If you're not incorporated, aren't you a sole trader?
Without these, I cannot account for the liability. The account shown on the invoice must be the company's account, i.e. not a private account, and I need the IBAN of the company account.
As a sole trader, your business account is in your name, obviously - check with your bank for the IBAN if you don't have that yet.
Our finance department have just told me that electronic bills are not yet accepted in Hungary,(...) First I've heard of this! (...)
Well, strictly speaking, that's exactly the situation in Germany, too - unless you have a certified electronic signature, electronic invoices are not eligible for tax purposes in Germany either.
My invoices ARE legal! I do not have a tax file number (VAT) and even if I did it would be in my personal name (as a freelancer) and definitely NOT in a registered company name!
But even if you're not VAT-registered, you need to have a tax registration number for your business - note that under recent EU rules, stating your tax number is mandatory.
What do you advise I do?
Issue a written invoice, stating your business details, IBAN/BIC and tax number.