Hi,
Yes I did reproduced the problem in your website, but not in my local website.
When (where) did I (or someone of the HikaShop support team) checked your shipping methods ?
When there is no shipping method selected and the option "auto select" is activated, HikaShop will recreate the data in the session to select a first available shipping method.
So, in your case, there is some data but the data is not understand by the rest of the process ; more than that, if you validate the checkout, you stay on the same step and there is no error message.
It is not logical regarding the source code of HikaShop, it should have something strange in your Joomla/HikaShop session.
So without more information I can't say more about your problem except that, I can reproduce it in your website for sure, but not in local.
Regards,