Maybe I'm blind but I can't find a way to access an organization that has granted me access via SSH key. I know it's a breeze connecting directly to your own account but how would I go about doing this through an organization?
Your account's personal SSH keys identify your GitHub user. If an organization has granted you access, it was done via your GitHub user; an SSH key attached to that user will give you access.
If you want to have separate SSH keys for your organization work (e.g. if you work on another machine), simply add the second key to your personal account.