Вы хотите метод onclick, здесь он передает ту же самую работу onclick.
handleSubmit(){
console.log(this.state.author);// u can see the value of autor and say somthing input vale in here it is coming right value or not
console.log(this.state.text);
axios.post(route,{
name:this.state.author, //these name and say are the variable whice are use to take the values to back end
Say :this.state.text
}).then({response}); //here u cn give some thing to disply after data add to database.
}
<form className="commentForm" onsubmit={this.handleSubmit.bind(this)}>
<input
type="text"
placeholder="Your name"
value={this.state.author}
onChange={this.handleAuthorChange}
/>
<input
type="text"
placeholder="Say something..."
value={this.state.text}
onChange={this.handleTextChange}
/>
<input type="submit" value="Submit" />
</form>