Waffle supports private repos from organization accounts within GitHub. Your organization's administrator needs to whitelist Waffle as a trusted third-party application. Instructions for doing so can be found from GitHub here.
Once Waffle is trusted for your organization, you can start using it. To add a repository, you'll need appropriate access permissions (this is managed from within GitHub). Log in to Waffle (which will authenticate with your GitHub account) and navigate to your repo's Waffle URL (the format is waffle.io/organization/repo).
Please note: once you add Waffle as a trusted application for your org, you must log out and re-authenticate on waffle.io.