Sending FBA Goods to EU without VAT registration?

No, in order to use FBA in a country in which you or your business are not resident, you must be VAT registered in that country.

Are you VAT registered in the UK?

Amazon VAT services can arrange VAT registration in any or all of the EU countries. Note:

You can only use the service if you are VAT registered in your home marketplace/country - they do not handle VAT in your domestic marketplace.

Promotional offers are for the first year only and will be expensive from the second year (depending on what you consider expensive of course)

Once you have committed to registering in your first countries I don’t believe Amazon VAT services will then process further registrations in other additional countries so decide what you want up front.

Note Note:

Everything will change in July - I believe that a One Stop registration may be available from July that would drastically reduce costs and simplify EU FBA selling from a single VAT registration but I don’t know much (more) about it. It would be worth looking into this before leaping into old-style registrations now.