How to add new field to the account address?

The other guys assumed that you wanted to add your own store’s VAT number, but I read it at you want the customer to add their VAT number when they create an account or order as guest.

This is certainly possible, and WooCommerce does have a whole mechanism to hook into adding additional customer account fields. However, it’s quite in-depth and not for the faint-hearted. I found this great tutorial that helped me get to grips with it. It’s quite long, but it does take you through every place where the additional field(s) need to be registered.

https://iconicwp.com/blog/the-ultimate-guide-to-adding-custom-woocommerce-user-account-fields/

Hope that helps