I am not sure I understand what does 'puma' organization means. AppVeyor Account in question is nateberkopec. I do not see any other AppVeyor URLs in the original message, which point to any other AppVeyor account. If you need to be able to start builds in that account, account owner should make you a collaborator.
I feel that here is a confusion between GitHub permission model and AppVeyor one. They are not toed, because AppVeyor supports many others version control systems like Bitbucket, GitLab e.t.c. This was the reason I sent this link in the original answer.
on 20 Mar, 2018 05:28 PM
The "puma" organization refers to http://github.com/puma/ it is not a
person, but instead an organization. I have commit access to this
organization. Nateberkopec is an individual, his personal projects are at
github.com/nateberkopec. If he gave me access to `nateberkopec` then I
would be able to modify any of his personal files and see his private
repos, which is not what either of us wants. The puma project does not
belong to `nateberkopec` on github, or it would be found at `
github.com/nateberkopec/puma` instead it is at `github.com/puma/puma`.
Nateberkopec set up the integration with appveyor for `github.com/puma/puma`
because he is a maintainer of github.com/puma/puma which is in the
github.com/puma organization. How do I set up the project so that everyone
who has access to github.com/puma/puma can restart builds on appveyor?