[Vuejs]-How to insert data to database with vue and laravel

that is typo error:

change addartist/store-artist to artist/store-artist in route

Your api is :

Route::group(['middleware' => 'cors'], function(){
    Route::post('addartist/store-artist', 'ArtistController@store');
});

and you are doing :

axios.post('/api/artist/store-artist', this.$data.artist)
                    .then((response) => {
                        alert('Success add Artist')
                        console.log(response)
                    })