You can sell VAT free to companies in other EU countries if they register as a customer in your shop and at registration provide their VAT ID.
The shop can then automatically deduct the VAT for that customer. You can set the VAT ID as visible in the registration form on the page Settings > Basket settings > Registration.
The VAT ID is made up of the country code followed by a number without any spaces or hyphens. A list with all the countries' algorithms can be found here.
You can create an instruction page about this in your shop where you can urge corporate customers to provide their VAT ID in order to get VAT free prices.