Can I claim a VAT exemption on orders from Printify?

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

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

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 Printify directly from the tax authority by including the relevant VAT invoices into your VAT declaration.