Sending emails from a website is always a giant pain. The current internet email environment has taken drastic measures to prevent spam, and as a result it's pretty hard to just send an email and expect it to get delivered.

A lot of times the emails are getting flagged as spam. Have the site owner check his spam box. If it has your emails then get him to un-spam them and go back to your setup when he was getting the emails. If they are totally blocked, you can also try one of the online services that check the spamminess of your emails. They will recommend you add dkim records and do other stuff, but a lot of times that's a moving target and still not a guarantee that your emails will go where they need to go.

Using Google or Office 365 can help, but they have rate limits on how many emails you can send per hour and if your site gets a lot of traffic this can become a bottleneck. I have had some success using SendGrid, but you have to pay for an account (https://github.com/sendgrid/smtpapi-php).