Hi,
I was able to find the issue.
It comes from three things independent from HikaShop:
1. As I expected, your Joomla template doesn't display error messages.
If I try again your checkout on the dev website with the default template, I get that error message:
monosnap.com/file/A8IFkp1EBoC4JHWAjcFS4TRItVU8o4
2. You have unpublished the "order_delivery_date" custom order field on which the "Brainforge Order Delivery Date" bases itself.
3. There is no check in that plugin to skip the verifications of the delivery date if the custom field is unpublished.
So you have 3 things you should do:
1. Check with your template provider why the error message is not displayed (it happened here, but it can happen in the future for other things and it's important to have error messages displayed when there is a real issue. For example, if the customer tries to use his email address to register while he already has a user account, you want to display the error message to tell him that the email is already used so that he knows he should try to login. Right now, since the error messages don't display, you can get such situation.
2. I suppose you want to either republish that custom field or disable the plugin using it.
3. You want to contact the developer of that plugin so that they can add a check to skip the delivery date field check if the custom field is unpublished.