UK businesses will eventually need to deal with border controls when goods are shipped across the UK-EU border. This could range from satisfying rules of origin to increases in import duties. This depends on which trading arrangement replaces EU membership.
The UK could negotiate a new trade agreement with the EU, similar to the bilateral agreements that Switzerland signed up to in 1999. Then it would be part of the European Economic Area (EEA), this would make things easier as there is free trade between EU and EEA members.
If Britain are not able to strike a trade deal with the EU, then import duties negotiated at the World Trade Organization would kick in.
In any case Britain would not have access to the coordinated VAT collection of the EU. When importing this would mean 20% VAT at the UK border, and the importer would no longer have the convenience of combining this with domestic VAT payments.
Right now, nothing will change. As it stands there is to be a 2-year exit negotiation period, during which EU laws will still apply to the UK.