Woocommerce in standard do not have VAT field on the customer. THis field is usually added either by some paid plugins or by custom development. Once this field exists on Woocommerce you need to check it’s technical name and add setting on the “Customer“ tab on the related integration
In the field “VAT/Reg. Number Field“ select name of the field in which you want to add downloaded VAT number. On screenshot above there is selected standard Odoo field “Tax ID“. Usually this field is used in Odoo standard for storing VAT number. But you can use any Char field if you want.