Can I claim a VAT exemption on orders from Printful?

No, in contrast to the US sales tax exemption system, the VAT system does not have exemptions based on the usage of the goods.

As Printful sells only taxable goods, all of your orders are subject to VAT.

The good news is that if you are registered as a VAT-payer in the relevant jurisdictions, you should be able to reclaim the VAT amount charged by Printful directly from the tax authority by including the relevant VAT invoices in your VAT declaration.