Brexit will with near certainty affect the way VAT is applied on cross border transactions between the UK and the remaining EU countries. The exact form this will take may take years to unravel.
Brexit may allow for greater flexibility in VAT policy design for the UK Government in future, but due to the complexity of application of VAT across EU member states it will not be straightforward.
There could be particular effect on how US companies with UK subsidiaries handle VAT on services revenue for their non UK based customers within Europe. There are quite stark differences on VAT treatments for different categories of charge, and it’s possible that there will be difference in the rules around place of supply.
Other changes may be in the way that non EU countries can claim back input VAT on suppliers.
That being said, claiming VAT on suppliers outside of the UK but elsewhere in the EU has never been quite as straightforward as the various agencies involved would like to advertise despite several EU VAT Directives mandating a certain degree of uniformity. In reality many EU countries continue to administer VAT reclaims at their own pace and generally speaking tend not to communicate in English.