If the language pair isn't fully selected, i.e., if there is not a Visitor and/or Agent language selected, the send button or message field will be greyed out and you will be prevented from sending a message through.
Make sure you select both languages at the top of the Unbabel Chat panel and try again.

The Unbabel component may also prevent you from sending a message if you lost connection to the chat. Make sure the connection to the chat is established and try again.