If you have just set up your Commerce, chances are that you have configured any SMTP to your WordPress site.

You need SMTP to manage emails coming it and out from your site. You need so that you will receive email from users who send message through the contact form. Anything that has to do with email you need an SMTP for that and if you have not configured anything then thats probably the reason your buyer is not receive the email.

This integration can be done with a plugin called WP Mail SMTP. This is actually good guide to why we need SMTP and how you could implement it.