If you are already VAT registered in your home country, then you won’t need to immediately register for VAT in the countries of your customers to sell to them. The European Union, which sets the overarching rules for VAT in the 28 member states, has set VAT registration thresholds for foreign FBA sellers. If you are selling below these annual thresholds in a particular country, then you don’t need to register there and you can charge the VAT under your own domestic VAT number.
You can check these distance selling VAT registration thresholds here.
Once you exceed the threshold in a particular country, you will have to VAT register in the target country as a non-resident VAT trader. Under the EU’s Single Market rules, this enables you to sell in a given EU state without the need to set up a local company. However, you still need to charge the local VAT rate and remit it to the local tax authorities because the EU has not harmonised its state tax regimes.
In addition to needing a VAT registration for exceeding the distance selling thresholds, you will also require one if you are storing goods in a foreign Amazon warehouse prior to sale to local consumers.