Braintree does not have native support for collecting Tax/VAT numbers from customers, however ChargeDesk can collect this data provided it is formatted as a custom field associated with the customer.
Custom fields must be added via your Braintree control panel before they can be used. Braintree provides documentation on how to setup a custom field on your Braintree account.
When creating the custom field, the API name should be set to tax_number and Store and Pass Back should be selected as the field type.
The Custom field creation form should look like this on your Braintree account. Once you have saved this field, you can then enable collecting VAT numbers on ChargeDesk checkout forms. You can also save tax numbers to this field in your own Braintree integration.