Hi @horchill. WordPress will, by default, send notification emails to the Administration Email Address when a new user registers.
Check that the Administration Email Address is correct by logging into the WordPress admin interface and navigating to Settings → General.
It could be that your WordPress installation is not able to send any emails at all. You can test this by doing something that results in an email being sent e.g. submitting a contact form or clicking “Forgot password” on the login screen.
If your site is not able to send any email, I recommend speaking with your hosting provider for help fixing this. Alternatively you can use a plugin like WP Mail SMTP to configure sending mail using a third party service.