Fetch post block/field data as JSON with Wordpress API

Very old question, but if anyone is stumbling around the Wordpress Rest API like I am, here's a quick snippet you can add to the functions.php file on your theme to add a blocks field on the return post/page data.

add_action(
    'rest_api_init',
    function () {

        if ( ! function_exists( 'use_block_editor_for_post_type' ) ) {
            require ABSPATH . 'wp-admin/includes/post.php';
        }

        // Surface all Gutenberg blocks in the WordPress REST API
        $post_types = get_post_types_by_support( [ 'editor' ] );
        foreach ( $post_types as $post_type ) {
            if ( use_block_editor_for_post_type( $post_type ) ) {
                register_rest_field(
                    $post_type,
                    'blocks',
                    [
                        'get_callback' => function ( array $post ) {
                            return parse_blocks( $post['content']['raw'] );
                        },
                    ]
                );
            }
        }
    }
);

Resulting in the following JSON being returned from fetching a single post:

"blocks": [
        {
            "blockName": "core\/paragraph",
            "attrs": [],
            "innerBlocks": [],
            "innerHTML": "\n<p>Here is the story content for section 1.<\/p>\n",
            "innerContent": [
                "\n<p>Here is the story content for section 1.<\/p>\n"
            ]
        },...

Same here, although this is old, it's useful, so here's another answer:

add_action('rest_api_init', function () {
  if (!function_exists('use_block_editor_for_post_type')) {
    require ABSPATH . 'wp-admin/includes/post.php';
  }

  // Get all post types that support the block editor
  $post_types = get_post_types_by_support(['editor']);

  foreach ($post_types as $post_type) {
    if (use_block_editor_for_post_type($post_type)) {
      register_rest_field(
        $post_type,
        'blocks',
        [
          'get_callback'    => function (array $post) {
            return parse_blocks($post['content']['raw']);
          },
          'update_callback' => function ($blocks, $post) {
            return update_post_blocks($post->ID], $blocks);
          },
          'schema' => [
            'description' => 'Gutenberg blocks as JSON',
            'type'        => 'array',
          ],
        ]
      );
    }
  }
});

/**
 * Updates a WordPress post with new Gutenberg blocks.
 *
 * @param int   $post_id The ID of the post to update.
 * @param array $blocks  The new blocks to insert.
 * @return bool|WP_Error True on success, WP_Error on failure.
 */
function update_post_blocks($post_id, $blocks)
{
  if (!current_user_can('edit_post', $post_id)) {
    return new WP_Error('rest_forbidden', __('You do not have permission to edit this post.'), ['status' => 403]);
  }

  if (!is_array($blocks)) {
    return new WP_Error('rest_invalid_param', __('Blocks must be an array.'), ['status' => 400]);
  }

  // Convert blocks back into serialized post_content
  $content = serialize_blocks($blocks);

  // Update post
  $result = wp_update_post([
    'ID'           => $post_id,
    'post_content' => $content,
  ], true);

  if (is_wp_error($result)) {
    return $result;
  }

  return true;
}