Please note your theme must include jQuery for the following code to work. You can grab a copy from https://cdnjs.com/libraries/jquery/ and add it to the top of your theme.liquid file, e.g.
<script src="insert copy of jQuery here"></script>
Within your cart.liquid or cart-template.liquid file, place the following code where you would like the delivery note field to appear
<textarea name="attributes[Delivery-Note]" style="display:none"></textarea>

Click Save

Open your storepickup-addons.liquid file and copy and paste the following code
<script type="text/javascript">
$(document).ready(function() {
window.ZapietEvent.listen('selected_method', function(method) {
if (method == "delivery") {
$('[name="attributes[Delivery-Note]"]').show();
} else {
$('[name="attributes[Delivery-Note]"]').hide().val("");
}
});
});
</script>

Click Save

Task complete! Now when a customer selects Local Delivery, an additional textfield will appear where they can add their custom delivery notes.