The store you're referring to is of version 4.1.6, that's an old version where there were no adding to cart without redirect (with ajax). The behaviour in question is still there, thus the feature is added as a custom mode.
If JS is enabled in the browser, and if the store considers that it's enabled, the JS script is supposed to send some data to script minicart_content.php, and the php script is to process the received data further. However it doesn't happen, and there are no JS errors, which makes me believe that the problem is in the code of minicart_content.php, with this file is very likely being modified too.
If JS is disabled in the browser ( and if you click the corresponding button in the store in the pink side menu block - "If Javascript is disabled in your browser click here"), this custom scenario is not applied, so the store uses the default functionality which allows to add the products to cart with out problems.
Thus, the possible solution is:
- roll back the custom changes you implemented, and use default functionality (adding products to cart with a redirect to cart page)
or
- check the minicart_content.php script and find out, why it doesn't want to properly process the data sent by JS script
If you're not sure how to achieve this, consider contacting X-Cart support team for further investigation. Not sure if I can give a link to X-Cart support here, but I'm sure you'll easily find it, if only try to search=)