how to insert content into my taxonomy

You’re using tax_input incorrectly. You’re not literally supposed to say taxonomy_name, you’re supposed to enter the name of your custom taxonomy. See https://codex.wordpress.org/Function_Reference/wp_insert_post. So, assuming your taxonomy is called ‘ninja_turtles’ and you are tagging your post with ‘donatello’ and ‘michelangelo’:

// Create post object
$my_post = array(
    'post_title' => 'My post',
    'post_content' => 'This is my post.',
    'post_status' => 'publish',
    'post_author' => 1,
    'tax_input' => array( 'ninja_turtles' => array( 'donatello', 'michelangelo' ) ) , 
    'post_category' => array(8,39),
);

// Insert the post into the database
wp_insert_post( $my_post );

For no particular reason (other than that it makes me feel like I have more granular control), I personally prefer not to use tax_input but to use the standalone function wp_set_post_terms() https://codex.wordpress.org/Function_Reference/wp_set_post_terms after the post is created:

// Create post object
$my_post = array(
    'post_title' => 'My post',
    'post_content' => 'This is my post.',
    'post_status' => 'publish',
    'post_author' => 1,
    'post_category' => array(8,39),
);

// Insert the post into the database
$post_id = wp_insert_post( $my_post );

// If all was successful, add the terms
if ( $post_id ) {
    wp_set_post_terms( $post_id, array( 'donatello', 'michelangelo' ), 'ninja_turtles' );
}