Will I have to pay taxes to ship to US address with a US passport?
Is this part of the VAT?
You are confusing two different things.
When you purchase just about anything in the EU, you pay VAT. It is not delineated in the price, the price shown includes it, though your receipt may break it out. You can, as a tourist, get a refund on that VAT if you are taking the items out of the country. That is a separate process of it's own, not sure if it applies to items shipped, or just carried out in luggage.
Packages coming into the US are subject to Customs, including any Duty owed. As part of the shipping process you add information for Customs (What is in the box, value, etc.) they will then notify you of any duty owed at delivery.