Email validation not working

Are other emails sent to the user ? If not, then, it's likely to be a problem of configuration of your emails settings in the joomla configuration.
If yes, could you give us the URL of your shop so that we can test that ?

I just tried to register a new account via the checkout on your website and I got the email.

I tried on your website and indeed had the problem you're describing.
It means that the problem comes from your email server rejecting the HikaShop activation email because of some setting in the Mail tab of the configuration of HikaShop. Please check these settings and try to change them and test if it works.
For example, some email servers, only accept a limited number of charsets, so you might want to try to change the charset of your hikashop emails to one of the charsets supported by your email server. You might want to ask your email provider what could be the problem in the mail configuration you have in HikaShop.

You have other solutions:
you could just turn off the activation of user accounts in joomla. That way, users will be able to continue the checkout without having to activate their account.
You could set the registration option to "no registration" in the tab checkout of the configuration of HikaShop. That will remove the joomla user account creation during the checkout.