You'll find the VAT ID of a customer in the following places.

The customer's note field:

On the order page of every order of a customer:

Note that the VAT ID may be displayed multiple times. This is because some apps (e. g. billing apps) expect the VAT ID to be in certain places and named certain ways and this varies from app to app. Because B2B Handsfree tries to support as many of these apps as possible, it stores the VAT ID in multiple places.
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