How to expose a custom field from sales_order table to rest api?

if you want to show this attribute in order collection as well you have to modify getList method also. Here is the complete solution:

 Create new column and set value for the existing order

Firstly, you need to create a new column in table sales_order in your installer file 

 $setup->getConnection()->addColumn($setup->getTable('sales_order'), 'delivery_type', 'varchar(1)');

In this post, we named it “delivery_type”.

etc/extension_attributes.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Api/etc/extension_attributes.xsd">
    <extension_attributes for="Magento\Sales\Api\Data\OrderInterface">
        <attribute code="delivery_type" type="string" />
    </extension_attributes>
</config>

etc/webapi_rest/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\Sales\Api\OrderRepositoryInterface">
        <plugin name="orderInformationUpdate" type="Vendor\Module\Plugin\Api\OrderRepository" />
    </type>
</config>

Plugin/Api/OrderRepository.php

<?php

namespace Vendor\Module\Plugin\api;

use Magento\Sales\Api\Data\OrderExtensionFactory;
use Magento\Sales\Api\Data\OrderInterface;
use Magento\Sales\Api\Data\OrderSearchResultInterface;
use Magento\Sales\Api\OrderRepositoryInterface;


class OrderRepository
{

    const DELIVERY_TYPE = 'delivery_type';

    /**
     * Order Extension Attributes Factory
     *
     * @var OrderExtensionFactory
     */
    protected $extensionFactory;

    /**
     * OrderRepositoryPlugin constructor
     *
     * @param OrderExtensionFactory $extensionFactory
     */
    public function __construct(OrderExtensionFactory $extensionFactory)
    {
        $this->extensionFactory = $extensionFactory;
    }

    /**
     * Add "delivery_type" extension attribute to order data object to make it accessible in API data
     *
     * @param OrderRepositoryInterface $subject
     * @param OrderInterface $order
     *
     * @return OrderInterface
     */
    public function afterGet(OrderRepositoryInterface $subject, OrderInterface $order)
    {
        $deliveryType = $order->getData(self::DELIVERY_TYPE);
        $extensionAttributes = $order->getExtensionAttributes();
        $extensionAttributes = $extensionAttributes ? $extensionAttributes : $this->extensionFactory->create();
        $extensionAttributes->setDeliveryType($deliveryType);
        $order->setExtensionAttributes($extensionAttributes);

        return $order;
    }

    /**
     * Add "delivery_type" extension attribute to order data object to make it accessible in API data
     *
     * @param OrderRepositoryInterface $subject
     * @param OrderSearchResultInterface $searchResult
     *
     * @return OrderSearchResultInterface
     */
    public function afterGetList(OrderRepositoryInterface $subject, OrderSearchResultInterface $searchResult)
    {
        $orders = $searchResult->getItems();

        foreach ($orders as &$order) {
            $deliveryType = $order->getData(self::DELIVERY_TYPE);
            $extensionAttributes = $order->getExtensionAttributes();
            $extensionAttributes = $extensionAttributes ? $extensionAttributes : $this->extensionFactory->create();
            $extensionAttributes->setDeliveryType($deliveryType);
            $order->setExtensionAttributes($extensionAttributes);
        }

        return $searchResult;
    }
}