I have done a test on the contact me form and I can't find where the emails are going to? Where is that configured on the site? I assumed it would go to the main admin email?
Emails are supposed to be sent to the email address you did set in Settings -> General.
If you are having issues with receiving emails, that might be due to an issue with your server. Please try installing this plugin: https://wordpress.org/plugins/wp-mail-smtp/