Connect API can be used to get Conversation Entries for a Messaging Session record in Salesforce Messaging for In-App and Web.

1. Use the following SOQL to fetch the Conversation Identifier of the Messaging Session record.

SELECT Conversation.ConversationIdentifier
FROM MessagingSession
WHERE Id = '<Id Of Messaging Session>'

2. Use the following path with your My Domain URL to fetch the Conversation Entries for Messaging Session in Salesforce Messaging for In-App and Web.

/services/data/v58.0/connect/conversation/<ConversationIdentifier>/entries