If you get a VAT number as soon as you incorporate your business in Estonia, then you can let your bank know when you open your account. If you already have a bank account, then you must tell them your VAT number.
Every bank is different and has a different process in place. If you do it with us and we help you open your bank account, we can take care of this task. If you do it yourself, your best option is to search your online banking account and see if you can either change it manually with a simple form or if you have to contact them via phone or e-mail.