Hi,
Even with the Essential or the Starter editions of HikaShop, emails are sent at the end of the purchase process.
If you don't receive the emails, there is a problem with either the email settings of the Joomla configuration or with the email settings of the HikaShop configuration which are not compatible with your email server.
The best in that case is to contact your hosting company in order to check those settings with them.
Otherwise, you can switch them on/off one by one, and test each time and see if that helps, but it's a long process and might not be enough to understand what's going on.
If you could provide a screenshot of these email settings, we could tell you what could likely be the problem, but I still recommend you to contact your hosting company.