This issue relates to Woocommerce settings. I am not sure 100%, however, I do not find the appropriate e-mail template in Woocommerce > Settings > Emails and I suspect that the default Woocommerce does not send email to admin, but to the client when a new order has been placed. If you do no receive emails at all, contact your host support.
Please check the issue on one of the default themes, try free Storefront to understand if our theme causes the problem. Our theme does not influence the e-mail functionality.