Law in Spain tells:
Your customers in Spain have to receive your VAT number in the invoices.
Article 6.1 d) Contenido de la factura (Invoice contents) del Reglamento por el que se regulan las obligaciones de facturación aprobado por Real Decreto 1619/2012, de 30 de noviembre (Executive rule for invoicing passed by bill 1619/2012):
Toda factura y sus copias contendrán los datos o requisitos que se citan a continuación, sin perjuicio de los que puedan resultar obligatorios a otros efectos y de la posibilidad de incluir cualesquiera otras menciones: [...] Número de Identificación Fiscal atribuido por la Administración tributaria española o, en su caso, por la de otro Estado miembro de la Unión Europea, con el que ha realizado la operación el obligado a expedir la factura. [...].
Invoices and copies will include these data or requirements, notwithstanding others with other purposes and the possibility of including whatever other mentions: [...] VAT number of the provider. [...]
Your customers in Spain have to receive their VAT number in the invoices in several cases, e.g. both you and them are in the EU (which would be true if you "are" in, say, Luxembourg, or the Netherlands, or Ireland (but funnily not if you "are" in several places of Spain such as Ceuta, Melilla or the Canary Islands), your lawyer can help you with this).
Article 6.1 d) Contenido de la factura (Invoice contents) del Reglamento por el que se regulan las obligaciones de facturación aprobado por Real Decreto 1619/2012, de 30 de noviembre (Executive rule for invoicing passed by bill 1619/2012):
[...] Asimismo, será obligatoria la consignación del Número de Identificación Fiscal del destinatario en los siguientes casos: 1.o [...] entrega de bienes destinados a otro Estado miembro [...] exenta [...]. 2.o [...] [inversión del sujeto pasivo] [...]. 3.o [...] operaciones que se entiendan realizadas en el territorio de aplicación del Impuesto y el empresario o profesional obligado a la expedición de la factura haya de considerarse establecido [...].
[...] Also, VAT number of the customer is mandatory when: 1.0 [...] goods shipments towards other member State of the [European] Union [...] exempt [...]. 2.o [...] [customer makes the invoice in behalf of provider] [...]. 3.o [...] operations performed in the land where [Spanish/European?] VAT can be demanded and provided that the entrepreneur or business owner or professional providing the goods or services is to be considered established [...].
I guess you might be in a case of 6.1 d) 3.o
, surely your lawyers would know better than me.
Beware I have no special qualification or proven successful record of law translation.
BONUS: In case of restaurant tickets and those sort of (less formal?) receipts, "factura simplificada" in the Law in Spain, your customers in Spain still have the right to claim their VAT number to be there (by article 7.2 of Reglamento por el que se regulan las obligaciones de facturación aprobado por Real Decreto 1619/2012, de 30 de noviembre (Executive rule for invoicing passed by bill 1619/2012)). If you deny such, article 24 of Reglamento por el que se regulan las obligaciones de facturación aprobado por Real Decreto 1619/2012, de 30 de noviembre (Executive rule for invoicing passed by bill 1619/2012) gives your customer the right to call arbitration (but that's a story for other day).
Hope that helped.
DISCLAIMER: This is not your lawyer, it's a vague (yet informed) entry level reference in order to guide you. Here's no responsibility for any damages caused. If on doubt, always choose to go to your lawyer.