How do I access a receipt sent to my private Gmail?

There are no 'private' chats here.

This is a user to user forum.

If you someone sent something to your gmail account, then you need to log into your gmail account to access it.

If you are having a problem getting into your gmail account, you need to contact google for support.

This is a Apple product support forum.

if it's in your private gmail account then you would access it using a webbrowser or mail client setup to receive gmail, and likely the receipt would maybe be a attached pdf file in the mail

iCloud Private relay is a feature of iCloud+ You must have an iCloud+ subscription.

iCloud Private relay works with Safari.

gmail is a google product/service. I don't know why you think there is some connection between gmail and iCloud private relay. There is not.

There is no such thing as "private relay gmail on iCloud". That phrase does not make sense.

About iCloud Private Relay - Apple Support

try

How to Access Gmail: Guide for Desktop, iPhone, or Android

Your google account has absolutely nothing to do with whether or not you have iCloud + or even an iPhone.

Google and Apple are two completely different companies.

If you have a problem with your gmail account, contact google for assistance.