If the UK leaves the EU without an agreement, the UK will stop being part of EU-wide VAT systems and all EU driven laws/systems will impact UK businesses.
Currently the Directive 2006/112/EC: The Principal VAT Directive, contains the detailed rules and the overall principles for applying VAT in all the EU Member States. Although it is mandatory where it applies, Member States must implement it in their own country by incorporating it within national law. EU law has direct effect ie if appropriately framed, confer rights on individuals which the courts of member states of the European Union are bound to recognise and enforce before the national courts.
Other EU laws impended in EU legislation include ‘Fiscal neutrality’ and ‘Effectiveness of rights’. Fiscal Neutrality means that the VAT should impact in the same way on anyone carrying on the same activity within the state, otherwise the tax would ‘distort competition’ and harm the proper working of a free market in goods and services. Effectiveness of rights means where the EU law gives a person rights, national law should not take those rights away.
Businesses involved in importing goods from the EU, exporting goods to the EU, supplying services to the EU, and interacting with EU VAT IT systems such as the VAT Mini One Stop Shop (MOSS), using TOMS will certainly be affected. HMRC has been engaging with such businesses esp. the travel industry to minimise any impact.
No one knows what the No Deal situation will be like – but I guess it would be a mini Deal with basic agreements on imports, exports, VAT etc.