Is it possible to install a private add-on (that isn't published in the GSuite Marketplace) in my organization's Gmail?
If so, how can I do that?
I went through all the settings under Apps Settings of the Google Admin console, but I couldn't find how to do it. I couldn't find any documentation either.
According to https://developers.google.com/gmail/add-ons/how-tos/publish to allow other users on your domain to install your add-on you should give them the deployment ID.
Publishing Gmail Add-ons
Publishing add-ons allows them to be used by other users. While a few add-ons have been made available in the G Suite Marketplace for users to install, it is not currently possible to publish your own add-on to the Marketplace.
In the meantime, you can install add-ons you have developed to test them. You can also install add-ons created by other users in your domain if they give you the add-on's deployment ID.
If you'd like your add-on to be considered for publication, please fill out this form to be notified as we begin accepting additional listings.
The instructions for installing unplublished add-ons are published on https://developers.google.com/gmail/add-ons/how-tos/install-unpublished
