You can add more input fields inside the default Shopify sign up form as described in this article:

https://shopify.dev/themes/customer-engagement/additional-customer-information

Or, follow these steps:

(1) Open your theme's page:

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(2) Click Actions > Edit code

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(3) Search for customers/register.liquid file:

(4) Add the following code inside:

<div class="vat">

<label for="vat">VAT ID</label>

<input id="vat" type="text" name="customer[note][vat]" />

</div>

(5) Save your changes

Result: