Thanks Brendan!
Ok, there's a couple of critical things about what the OP has said here, and obviously it's in no doubt that they have perpetrated a fraud here.
They've asked how this might come to Revenue's attention. Well for starters, it seems they went through the process of ordering a TV and at the very end when presented with the opportunity to do so, they entered a VAT number. If that's an accurate version of what happened, they'll be bang to rights, as their name and billing address for payment by card will be linked to the transaction, notwithstanding the VAT number being linked to a completely different name and address.
The UK trader may be obliged to file a periodic return detailing their intracommunity sales to other registered traders, so the fact that the unwitting Irish trader has had their VAT number used will exist in a record that is shared with Irish Revenue.
Likewise, Irish VAT registered traders are supposed to include in their own periodic returns, the value of goods / services acquired from other VAT registered traders abroad.
If the trader whose number has been hijacked doesn't normally acquire goods from the UK, then there will likely be a mismatch whereby the system will show that they have made UK acquisitions (as reported by the vendor) but haven't reported any UK acquisitions themselves.
The scale of this transaction means that it's highly unlikely it would trigger a Revenue inquiry, but if for whatever reason the Irish trader has their VAT position for the period in question looked at by Revenue, then there is a quite high likelihood the intra-EU mismatch gets checked (because any vat inspector worth their salt will be all over purchases from the likes of Amazon, that aren't returned at the trader's side as clearly being likely to be personal stuff), in which case there's a clear paper trail straight back to the OP.