According to Github,
Support for SSH certificate authorities is available with GitHub Enterprise Cloud and GitHub Enterprise Server 2.19+. For more information, see "GitHub's products."
So if your organization account is not or cannot be subscribed to Enterprise plan (like if you're staying with either free or team plan) then you need to bypass github account as @Chris's answer suggests. (You can do work-arounds making new dedicated Github user for organization, but it complicates security concern as organization's administrator have to take care for that, so maybe that approach is not a wise one.)
If another platform is a choice, bitbucket allows you to do this for free per repository, not user. I use this on my Kiosk running on RaspberryPi so it to pull the update directly from repo. (My team is moving to Github's team
plan, so I'm torn apart in between changing deployment scheme to standard ways or keep this one repo on bitbucket for this one purpose.)
For gitlab though, it seems like it hasn't implemented this yet. (Link to issued request.)