You may definitely code your individual buying cart – however in the end that code ought to reside inside a customized plugin (or plugin performance housed in a customized theme). So the quick reply is “no” – a technique or one other it’s essential write or use a plugin.
Please be aware that plugin suggestions are off-topic on this stack. However WooCommerce is now an Automattic property – it’s developed and supported by company stewards of WordPress. It is pretty sturdy, however in case you have been coding your individual cart it’s possible you’ll discover that it’s too heavy to your wants.