The VAT rules are:
They charge VAT in their home country (Luxembourg), unless their sales to the EU country in question are above the distance selling threshold for that country, which can be set at either €100,000 or €35,000, or the approximate equivalent in local currency. It is £70,000 for the UK and Isle of Man. For digital goods (apps, kindle books, mp3s etc), they must charge VAT in the customer's country regardless of sales level, and it is the same for goods subject to excise duty such as alcohol and tobacco, though I don't think Amazon sells them.
As Amazon likely sells more than £70,000 of stuff every hour, never mind every year, they do have to charge UK VAT.