Can customers get order updates on WhatsApp?

Yes, with the WhatsApp Notification plugin. On the plugin's page:

1. Connect your Meta WhatsApp Business (Cloud API) account with its API token and Phone Number ID, and turn on Enable WhatsApp notifications.
2. Switch on each order event you want to send (such as placed, confirmed, shipped, delivered and cancelled) and enter its approved WhatsApp template name.

For browser and app alerts too, add the Push Notifications plugin.