Go to Settings > Store Info
It's important to determine what sending email address you are using. It's likely the emails are sending, but are being flagged/rejected by the recipient's email. AOL, Yahoo and vanity emails are known services that flag/reject. This can generally be avoided by implementing a Gmail fwd-ing address in the Sending Email field.
Create a new Gmail account
You will need to insert a Gmail address into the Sending Email field. Create a new Gmail account by going to Gmail.com and registering.
Important note: If you have an existing Gmail address, you are more than welcome to use that instead of creating a new one. However, you must set up forwarding on the account using the next steps.
Set up email forwarding within Gmail
Please visit Google's article on how to set up email forwarding by clicking here or following instructions below.
NOTE: Insert your shop's email (the one you are primarily using) as the email to forward to.
Insert your new Gmail address under Settings > Store Info
Make sure you insert your new Gmail as your sending address. You can test by sending a receipt to your personal email address. You should receive in either your inbox or, sometimes, Spam folder. If you reply to the email, it'll be sent to the Gmail account, then immediately forwarded to your primary business email (whatever you set up in your Gmail forwarding options)