Disclaimer: my business is based in the U.S. with little experience in the UK and EU. But I'm inclined to believe that if you are the entity legally importing the merchandise, you pay VAT. Your client would pay VAT only if the import documents list their name as the consignee. If you only wanted to charge a service of clearing customs and taking possession at the port without VAT liability, you would probably need to be licensed as a customs broker.

Again, I'm not an expert in clearance. A customs broker could better explain it, but I'd bet dollars to donuts I'm on the right track.

Thanks for your question!