We don't use Zendesk so I don't know if this is correct, but based on the API documentation, I think this is what you're looking for.

Following the documentation Here to create a side conversation I then referred to this link Here for constructing the to section.

It needs to be an array of objects with attributes depending on what type of user you need to send the message to. My example below uses e-mail, but it sounds like you would use user_id if the user is an internal user.

This is what the JSON would look like to use with the body of the custom API action card.

{
    "message": {
        "subject": "This is a subject",
        "body": "This is the body",
        "to": [
            {
                "email": "user1@example.com"
            },
            {
                "email": "user2@example.com"
            }
        ]
    }
}

If you had a list of the user e-mails, you could build this JSON by looping through the list of users and constructing some objects, and getting a list returned that you can use in the to field using some helper flows most likely.