Similar to EU-established companies, entities from outside the EU should have a legitimate business and should be registered taxpayers in their country of origin. 

They will be required to provide a Certificate of Incorporation indicating the name, address, legal status and legal representative of the company, a tax certificate, a passport copy of the legal representative, a bank statement and an Articles of Association (or similar document which specifies the regulations of the company’s operations and defines the company’s purpose). 
In case of partnership, a passport copy of the second person is also needed. 

In addition, Non-EU companies are required to appoint a fiscal representative in more than half of the 27 EU member states. For more information about Fiscal Representation please see separate article on the topic.

For country specific VAT registration document requirements please click below:

Sellers from China here
Sellers from India here
Sellers from other International countries here
Sellers from Turkey here
Sellers from USA 
here