I have found two problems with the current VAT Number implementation.
1) The web service which you use does not tell you if the VAT Number is valid it tells if the Company with the VAT number is registered as payer of VAT. There are companies which have VAT number but they are not VAT payers. VAT can be exempt to you if you are VAT payer from different EU country. Even if you are not VAT payer your VAT number should be included on invoice (at least according to Czech law). So, it means this needs to be changed:
VatNumberStatus.Invalid -> VatNumberStatus.NotVatPayer
VatNumberStatus.Valid -> VatNumberStatus.ValidVatPayer
2) It is common to enter VAT Number with a country prefix in Czech republic, customer are not used to enter it without it. I would add a drop down box with the prefixes before the VAT number. There can be connection between this drop down box and the drop down box with countries (if you change one the other is changed too).
It would be nice if entering a VAT number would fill company name, address and zip code. The web service returns this info. I would execute the web service right after entering the number. If it would be known VAT payer the details can be filled (but it should not overwrite details which are already filled). And the info about VAT Number status can be shown immediately too.