In order to change your VAT ID or to add a new VAT ID in your Amazon Seller central account you just have to follow the steps below:

Step 1.) Go to Settings / Tax Settings:

Step 2.) click on Add VAT numbers:

Step 3.) Click on Add a VAT/GST registration number:

Step 4.) Select the Country of the VAT ID, enter the VAT ID and select the registered address of the VAT ID: