Even though there is no option to hide the "Send attachment" icon by default, you are able to hide it using Custom CSS. To do so, navigate to Configuration > Chat > Chat buttons > Edit your chat button. Then in the "Chat window" tab scroll down to the "Custom CSS" section and add the following CSS code:
.MessageFormAttach {
visibility: hidden !important;
}
Do not forget to APPLY the CSS code and SAVE the chat window.