My website is live and the order confirmation emails are neither being sent to the customer nor the admin. I am using Magento 1.9.3.2.
Please check whether Cron jobs are working or not, you can check easily by installing AOE SCHEDULER extension, make sure you have proper setting in System > Configuration > Advanced > System > Cron
Also check whether you entered sender email in System > configuration> Sales> Sales email > Order emails.