You can opt into the European VAT by going to My Account->Edit Account (direct link is: https://www.jvzoo.com/myaccount/edit ). That will bring you to the various options that you can set for you account.
You can opt into the tax using the dropdown menu, which will then open the VAT options: http://screencast.com/t/e1xyQ48jhpwq
You will see that there are 2 sets of options. The first option is who will pay the VAT. The tax can be added to the price of the product, in this case the customer would have to pay the tax. The other option is for the vendor to pay the tax out of their profits from sales. The other option is how the tax is charged. Prior to 2015, VAT could be charged based on the vendor's country. As of 2015, the new laws state that the buyer's country should be used to determine the VAT tax rate. You can choose the rate in the second section.
The VAT payments to the government are completely up to the vendor. It is their responsibility to ensure that the taxes are paid at the correct rate. To that end, the vendor can get a breakdown of how much is owed by purchasing country.
You can find a breakdown of exactly how much the taxes are by going to the Sellers->Reports->VAT by Country page (located here: https://www.jvzoo.com/reporting/taxes/vatreports/ByCountry ):