Hi,
It's possible that your Joomla template is not displaying the error message:
www.hikashop.com/support/documentation/6...issues.html#messages
So I would recommend to try with the default template of Joomla and see if you get a proper error message.
For example, it could be that you set a weight in your products or you have the "force shipping regardless of weight" setting turned on in the Hikashop configuration. And if you didn't configure any shipping methods, then the checkout won't let you finish it as it requires a shipping method. In that case, you should see an error message, but your template might not be displaying it.
Otherwise, please provide precise instructions to reproduce the problem. I don't know where to go on your website to add a product to the cart and access the checkout. Also, please provide a screenshot of the settings under the Checkout tab of the HikaShop configuration page.