you should first verify how emails are configured to be sent in your instance.
Do you know if you are using the “mail” function (PHP) built within WordPress, or you have a dedicated SMTP solution to manage your “transactional emails” (all emails relating to the various stages of processing orders).
If you are using the built-in “mail” function, then your setup is wrong. It is well documented that this solution was not designed to handle transactional email content. Google search will return many articles on how to configure SMTP mail services via plugins, or otherwise for your site. If you are already using SMTP, then you should be able to test if your SMTP configuration is working or not.
If its, working then you move onto the next step of identifying if the email account you are using for the SMTP function is suitable for this purpose. Most general use mailboxes are limited to a maximum number of “X” emails per 24hrs. You should establish if you are hitting this limit, in which case you need to look into dedicated SMTP service providers for transactional emails. Once again Google is your friend, you will find many articles and options on transactional SMTP solutions, just be aware most will want to get paid something for their service, but many offer free services up to a number of emails per day to help a start-up business get established.
Been there, had to do allot of reading, sounds like you need to do so too ??