Hi,
Looking at the issue on your website, it seems to be linked to the popup system which doesn't work properly on your frontend.
So there are several ways you can try to solve the problem:
1. Change the popup mode setting under the Display tab of the HikaShop configuration. "vex" should be the best regarding compatibility issues but you can also try the other choices.
2. Change the address selector setting under the Checkout tab of the HikaShop configuration. If you don't use the popup selector, it should circumvent the problem.
3. You can also turn off the "checkout legacy" setting under the Advanced tab in order to use the new checkout instead of the legacy one you're using.