On the checkout page, by default, the Postal Code are required field.
vendor/magento/module-checkout/view/frontend/layout/checkout_index_index.xml
<item name="postcode" xsi:type="array">
<!-- post-code field has custom UI component -->
<item name="component" xsi:type="string">Magento_Ui/js/form/element/post-code</item>
<item name="validation" xsi:type="array">
<item name="required-entry" xsi:type="string">true</item>
</item>
</item>

So, you should check this field in your custom layout. Make sure the Required validation is enabled.
Additionally, if you want to make this attribute to be required on the server side also, you can use a simple query to :
UPDATE eav_attribute a SET a.`is_required` = 1 WHERE a.`attribute_code` = 'postcode';