Change <button type="button" to <button type="submit". Remove the onClick. Instead do <form className="commentForm" onSubmit={this.onFormSubmit}>. This should catch clicking the button and pressing the return key.
const onFormSubmit = e => {
e.preventDefault();
// send state to server with e.g. `window.fetch`
}
...
<form onSubmit={onFormSubmit}>
...
<button type="submit">Submit</button>
</form>