How can i set a default vat amount for a customer. i.e. EU customers to automatically be charged no vat?

Welcome to QuickBooks, 

You can not  set a default tax rate for a specific customer however you can set one for all transactions you create.  NO VAT would be the default setting for all transactions(expenses,bills, invoices etc)Let me walk you through with the steps.

1. Click the Gear icon in the upper-right corner and select Account and Settings.
2. Select the Advanced tab and go to the Accounting section.
3. Choose Inclusive of Tax in the Default tax rate selection field.
4. Click Save and Done.

Once done, you can select a tax rate. Moving forward, this will automatically set your default tax rate every time you create your sales forms. You will see the No Vat option. I've attached a screenshot for your guide.

If you want the NO VAT to auto apply but only for certain items on the sales receipt /invoice you can  set up chart of accounts to have a default tax rate, so for example if you were to set up a specific chart of account that you were going to use for these transactions you can set a default rate of NO VAT when you set up the chart of account itself. You can also do this when setting up a product or service. So that when you put that item in the invoice it will auto apply the NO VAT . 

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