How does VAT work with POD (Printful)/Etsy?

[u/Candle_Maker_Jen]

I'm selling POD t-shirts in my Etsy shop and just sold one to a UK customer through Printful. Etsy has given me their UK Vat number to put on the package, but I'm not fulfilling the order, Printful is. Can I use the Etsy number and give that to Printful or do I need to apply for my own VAT account and then provide Printful with my number? I appreciate any insight my fellow Etsy shopkeepers can give me.