Consult a tax specialist to learn about your tax obligations and whether you're required to charge VAT.
Shopify plans to simplify its tax settings for international customers. As of now, it's possible to set up tax for the US and the EU, but it's a tricky process.
One option is to create two separate stores: one that caters to North American customers, and another that caters to the EU. You could enable a redirect to your European store, or create two domains – .com and .co.uk.
Or you could create separate shipping territories (US and the EU) and configure taxes for each territory. Here's how:
Before you set VAT, you need to add the European Union as a shipping zone. Go to Settings → Shipping → Add shipping zone (or edit your current zones so the EU is separate)
Enter the zone name as European Union and select each country in the European Union
Enter a shipping rate
After you set your EU shipping zone, go to Settings → Taxes
Add your VAT rate to each country. The VAT rate you charge must be the same rate for the the country in which you're registered as a VAT payer.
Since VAT is also applied to shipping, be sure to check the “Charge taxes on shipping rates” box. This won't affect how sales tax is applied to US shipping.
Starting October 1st 2017, Printful will charge 21% VAT for orders fulfilled in Europe with a shipping address in the EU.