To update your Facebook page for Messenger communication, go to Settings and navigate to Messenger Platform.

Facebook page settings - Messenger Platform

Add your domains and the ManyChat domain to whitelist them.

whitelist domains on facebook page settings

Next, navigate to General to update the Messages setting that allows people to contact you via Messenger.

Update the Messages setting under the General section.

Check this box to allow people to contact you via Messenger.

Check the Messages checkbox to allow Messenger communication