This will require Apple to add support - currently, apps have no control over private relay.
Private relay can be used currently for Safari, Safari Tech Preview, and other apps that make plain unsecure HTTP requests (i.e. port 80), as well as for the built-in mail app. There is a whitelist of bundles currently in place, so I don't think there's anything third party apps can do currently.
Perhaps when it leaves private beta, there will be options for other apps to use it. With WWDC coming soon, it might be worth looking to see what comes up.