You can certainly code your own shopping cart – but ultimately that code should live inside a custom plugin (or plugin functionality housed in a custom theme). So the short answer is “no” – one way or another you need to write or use a plugin.
Please note that plugin recommendations are off-topic on this stack. But WooCommerce is now an Automattic property – it is developed and supported by corporate stewards of WordPress. It’s fairly robust, but if you have been coding your own cart you may find that it is too heavy for your needs.