Hi,
Sorry for the long message, but email issues require that.
First, check the email settings of the Joomla configuration page. They must be configured accordingly to your hosting provider recommendations so that your website can send emails.
You can use the "send test email" button there to check if your website is able to send emails.
If that doesn't, nothing on your website will be able to send emails, including the registration form, HikaShop and other extensions.
So you'll have to check with your hosting what is missing.
Once, the "send test email" is working and you're receiving the test email, if you still have the problem with the activation email, then :
Note that there are two registration forms, HikaShop's and Joomla's. For Joomla's, HikaShop is not involved. However, if the test email is working in the configuration, then it should work too. One thing I can say is that the registration settings of Joomla are available via the "options" button of the Joomla user manager. So you can check them. There is one to activate / deactivate the activation email. So deactivating it would allow you and your users to register on the website without having to use the verification code / link. And this also goes for HikaShop's registration.
And regarding HikaShop's registration form, the email sent is called "user account" and can be configured via the System>Emails menu. There, if you disable the email, then it won't be sent by HikaShop. Also, once an email is sent by HikaShop, it is logged in the Customers>Emails history menu. So, you can go there and check the emails being sent. If the email appears there but is not received by the receiver, it's likely there is a problem with the email once it is being processed by the email server of your hosting or the receiver email server. Normally, the reason for an email being lost by that point is logged in the error log of the email server of the hosting. So your hosting support should be able to provide you with information on what is missing in such a case. Usually, it's because the email is flagged as spam and you can use
www.mail-tester.com/
to check the spammyness of your emails and get directions on what to do.