const options = {
  method: 'PATCH',
  headers: {
    'x-spree-api-key': '<api-key>',
    Authorization: 'Bearer <token>',
    'Content-Type': 'application/json'
  },
  body: JSON.stringify({kind: 'eu_vat', value: 'DE123456789'})
};

fetch('http://{defaultHost}/api/v3/store/carts/{cart_id}/tax_identifier', options)
  .then(res => res.json())
  .then(res => console.log(res))
  .catch(err => console.error(err));

curl --request PATCH \
  --url http://{defaultHost}/api/v3/store/carts/{cart_id}/tax_identifier \
  --header 'Authorization: Bearer <token>' \
  --header 'Content-Type: application/json' \
  --header 'x-spree-api-key: <api-key>' \
  --data '
{
  "kind": "eu_vat",
  "value": "DE123456789"
}
'

{
  "id": "txi_UkLWZg9DAJ",
  "kind": "eu_vat",
  "value": "DE100000008"
}