Do I need a privacy policy to take orders on WhatsApp?

Yes, on the business API. WhatsApp's business policy says you must secure the notices, permissions and consents you need, including maintaining a published privacy policy.

• The policy says you are responsible for notices, permissions and consents to collect, use and share people's information.
• That includes maintaining a published privacy policy and following the law that applies to you.
• You may not use data about a person you message for anything beyond what is needed to support messaging with them.
• Keep the policy short, put it on a page you own and link to it.