Why is VAT an issue?
For cross-border goods transactions, there is a key difference between movements of goods intra-EU, and the movement of goods in and out of the EU.
Movement of goods between EU member states (intra-EU)
Where goods are situated in one EU member state, and sold and transported to a VAT registered businesses in another EU member, we call them dispatches (movements out of member states) and acquisitions (movements into member states). For example, if you are a UK VAT registered business, and buy machinery from Ireland, and it is sent to you in the UK, your Irish supplier has dispatched the machinery, and you have acquired it.
The Irish supplier charges 0% Irish VAT on the dispatch, and the goods move freely to you in the UK. There are no border checks - the goods of course comply with UK regulations because the Irish regulations are the same (common EU rules, and all that). No VAT is paid at the border, and the machinery happily arrives at your office.
Does this mean machinery is VAT free? Ha, no. But, the UK business self-accounts for VAT (acquisition VAT) on its UK VAT return at whatever rate prevails in the UK (20% for most machinery). HMRC has a way of checking compliance with this - the EU-wide Intrastat reporting system. (The Irish supplier must declare the goods sold to the UK business on its Intrastat report, and the Irish authorities share this report with HMRC).
Movement of goods in and out of the EU
Movements of goods in and out of the EU are imports and exports. If the UK now acquires the same machinery from a US supplier, it is exported from the US and imported into the UK.
The US does not have the same Intrastat reporting system (or, actually, VAT in general) so there is no way of HMRC checking what VAT needs to be paid by UK suppliers on these imports. As such, the goods are stopped at border, and import VAT is paid before the goods can clear Customs (alongside Import Duty and other regulatory requirements in many cases). This means there is currently a need for a hard border between EU and non-EU countries for VAT purposes, regardless of other issues.