Magento 2.3 – Pass Custom Input Field Data to Sidebar Summary

you need to create a plugin in your custom module ( the one that you used to create the custom attribute ), and have the code something like the following one:

namespace Package\Name\Plugin\Checkout;

class LayoutProcessor
{
    public function afterProcess(
        \Magento\Checkout\Block\Checkout\LayoutProcessor $subject,
        array $jsLayout
    ) {
        // Loop all payment methods (because billing address is appended to the payments)
        $configuration = $jsLayout['components']['checkout']['children']['steps']['children']['billing-step']['children']['payment']['children']['payments-list']['children'];
        foreach ($configuration as $paymentGroup => $groupConfig) {
            if (isset($groupConfig['component']) AND $groupConfig['component'] === 'Magento_Checkout/js/view/billing-address') {

                $jsLayout['components']['checkout']['children']['steps']['children']['billing-step']['children']
                ['payment']['children']['payments-list']['children'][$paymentGroup]['children']['form-fields']['children']['custom_attribute_code'] = [
                    'component' => 'Magento_Ui/js/form/element/abstract',
                    'config' => [
                        'template' => 'ui/form/field',
                        'elementTmpl' => 'ui/form/element/input',
                        'id' => 'custom_attribute_id',
                    ],
                    'dataScope' => $groupConfig['dataScopePrefix'] . '.custom_attribute_id',
                    'label' => __('Custom attribute label'),
                    'provider' => 'checkoutProvider',
                    'visible' => true,
                    'validation' => [
                        'required-entry' => true,
                        'min_text_length' => 0,
                    ],
                    'sortOrder' => 300,
                    'id' => 'custom_attribute_id'
                ];
            }
        }

        return $jsLayout;
    }
}

Hope this helps

Based on your question I'm under the assumption you already have your extension attributes set up. I have carried out a similar modification and hopefully my answer helps.

In your Custom module create a requirejs-config file to extend the default shipping processor/default

Namespace/CustomModule/view/frontend/requirejs-config.js
var config = {
    "map": {
        "*": {
            'Magento_Checkout/js/model/shipping-save-processor/default': 'Namespace_CustomModule/js/model/shipping-save-processor/default'
        }
    }
};

Add your extension attribute to the payload.

/*global define,alert*/
define(
    [
        'jquery',
        'ko',
        'Magento_Checkout/js/model/quote',
        'Magento_Checkout/js/model/resource-url-manager',
        'mage/storage',
        'Magento_Checkout/js/model/payment-service',
        'Magento_Checkout/js/model/payment/method-converter',
        'Magento_Checkout/js/model/error-processor',
        'Magento_Checkout/js/model/full-screen-loader',
        'Magento_Checkout/js/action/select-billing-address'
    ],
    function (
        $,
        ko,
        quote,
        resourceUrlManager,
        storage,
        paymentService,
        methodConverter,
        errorProcessor,
        fullScreenLoader,
        selectBillingAddressAction
    ) {
        'use strict';

        return {
            saveShippingInformation: function () {
                var payload;

                if (!quote.billingAddress()) {
                    selectBillingAddressAction(quote.shippingAddress());
                }
                //Adding the extension attributes to your shipping address
                payload = {
                    addressInformation: {
                        shipping_address: quote.shippingAddress(),
                        billing_address: quote.billingAddress(),
                        shipping_method_code: quote.shippingMethod().method_code,
                        shipping_carrier_code: quote.shippingMethod().carrier_code,
                        extension_attributes:{
                            custom_field: $('#custom_field').val(), 
                        }
                    }
                };

                fullScreenLoader.startLoader();

                return storage.post(
                    resourceUrlManager.getUrlForSetShippingInformation(quote),
                    JSON.stringify(payload)
                ).done(
                    function (response) {
                        quote.setTotals(response.totals);
                        paymentService.setPaymentMethods(methodConverter(response.payment_methods));
                        fullScreenLoader.stopLoader();
                    }
                ).fail(
                    function (response) {
                        errorProcessor.process(response);
                        fullScreenLoader.stopLoader();
                    }
                );
            }
        };
    }
);

Save the Attribute to your Quote with a plugin (Not sure if you could use an observer here I have not checked).

di.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
    <type name="Magento\Checkout\Model\ShippingInformationManagement">
        <plugin name="Namespace_CustomModule_save_delivery_date_in_quote" type="Namespace\CustomModule\Plugin\Checkout\SaveAddressInformation" />
    </type>

</config>

SaveAddressInformation.php

class SaveAddressInformation
{
    protected $quoteRepository;
    public function __construct(
        \Magento\Quote\Model\QuoteRepository $quoteRepository
    ) {
        $this->quoteRepository = $quoteRepository;
    }
    /**
     * @param \Magento\Checkout\Model\ShippingInformationManagement $subject
     * @param $cartId
     * @param \Magento\Checkout\Api\Data\ShippingInformationInterface $addressInformation
     */
    public function beforeSaveAddressInformation(
        \Magento\Checkout\Model\ShippingInformationManagement $subject,
        $cartId,
        \Magento\Checkout\Api\Data\ShippingInformationInterface $addressInformation
    ) {
        $extensionAttributes = $addressInformation->getExtensionAttributes();
        $customField = $extensionAttributes->getCustomField();
        $quote = $this->quoteRepository->getActive($cartId);
        $quote->setCustomField($customField);

    }
}

Save the Attribute to your order with an Observer events.xml

<?xml version="1.0" encoding="UTF-8"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Event/etc/events.xsd">
    <event name="sales_model_service_quote_submit_before">
        <observer name="unique_observer_name" instance="Namespace\CustomModule\Observer\SaveCustomFieldToOrder"/>
    </event>
</config>

SaveCustomFieldToOrder.php

class SaveCustomFieldToOrder implements ObserverInterface
{
    /**
     * @var \Magento\Framework\ObjectManagerInterface
     */
    protected $_objectManager;

    /**
     * @param \Magento\Framework\ObjectManagerInterface $objectmanager
     */
    public function __construct(\Magento\Framework\ObjectManagerInterface $objectmanager)
    {
        $this->_objectManager = $objectmanager;
    }

    public function execute(EventObserver $observer)
    {
        $order = $observer->getOrder();
        $quoteRepository = $this->_objectManager->create('Magento\Quote\Model\QuoteRepository');
        /** @var \Magento\Quote\Model\Quote $quote */
        $quote = $quoteRepository->get($order->getQuoteId());
        $order->setCustomField( $quote->getCustomField() );

        return $this;
    }
}