If your customers are not receiving email notifications from you, then it's most likely that the emails are either in a spam folder or bouncing back, which in turn prevents the customers from receiving it.
First, I would definitely recommend asking the customers to check their spam folders to see if the email got lost there.
If there's none, then adding Shopify's SPF record should be able to help resolve this.
You can learn more about this topic in the link below:
Shopify Help Center (See "Add Shopify's SPF record to your domain host (custom domains only)" for the resolution)
You will need to access your domain's DNS settings in order to solve this. If you're using a third-party domain, then only you can access it. If you're using a Shopify bought domain, then this can be done via the Shopify Admin > Online Store > Domains.
Doing this is just a few clicks away. :) Here's a guide on how you could do this:
Log in to your domain hosting account.
Find your domain's DNS settings.
Then, add a new TXT record (or find your domain's TXT record) with the following contents:
Under TXT 'Name', type in: @
Under 'Value', type in: v=spf1 include:shops.shopify.com ~all
*Please note that after you update the SPF record, it can take up to 48 hours for the changes to take effect.*