Au contraire, ils me demandent bien une facture. Mais pour eux le n° de TVA est indispensable.

in order to operate with companies and people in other european countries the EU obliged me (everybody) to have an Intracommunity VAT number and to ask my suppliers to have one too, precisely to avoid double taxation and VAT (VAT is different in any country).

This is an useful link: VIES (VAT Information Exchange System) enquiries - European commission
Et ils me fournissent un exemple de facture d'un autre photographe, anglais celui-là.