I don't believe the EU can mandate tax collection on behalf of EU Countries to companies outside EU boundaries. You are only compelled to follow the laws of your own Country, which include all International treaties your Country has entered into. You should research what your Country mandates about this subject matter.

The issue is problematic in the following instances.
In the UK for example companies and traders are not liable to charge VAT on sales unless.

1 ) they have registered and have a GB xxx-xxxx-xx VAT number for accounting.
2 ) UK import (purchasing goods and services) may be subject to different levels of VAT. Typically items for sale in educational environments are zero rated.
3 ) liability for charging VAT in a the UK is subject to a turnover in access of 81,000GBP.

Thus as a trader having a turnover of less than 81,000 GBP need not register, and charge VAT. Remember that VAT is designed to be payable on Value added, this means that Input tax can be used to offset output tax.

Example,
I buy an item for 100.00 the VAT is 20% = 120.00 I then sell the item for 120.00 excluding VAT, the sale price will be 144.00. So the input VAT is 20.00 and the output VAT is 24.00 so the VAT due revenue is 4.00 (the output minus the input).

The regular consumer cannot recover the final VAT, which is accumulating as the profit increases.

There is a further issue. VAT is chargeable on Imports, and this is where the EU policy is trying to collect revenue. This is determined on import for physical goods and services. This is ordinarily based on the invoice value and subject to Duty and VAT. There are exclusions as noted above, plus a few others. Even if the companies outside the EU collect VAT it seems unlikely that the authorities would have any jurisdiction to collect the monies from let's say China. I doubt whether Chinese companies would volunteer payments as there is very little chance that the UK Government would authorise their staff to go on world trips to recover the dubious amounts. In some cases, typcally low value items imports are not charged VAT.
Imports from outside the EU are treated differently:

. you must make an import declaration to customs
. you generally have to pay import duty and import VAT (plus VAT on import duty), although use of some customs procedures may suspend or relieve you from these taxes
I work in the Freight industry and deal with physical imports and exports into the UK.