First off, TubeBuddy is not compatible with CMS accounts on YouTube.
If you have granted someone access to your YouTube channel through YouTube Studio (here: https://www.screencast.com/t/7VqJiwNn) then it's going to take an extra step in order for them to use TubeBuddy on your channel.
Note that if you switch from YouTube's channel management system to Google's channel management system, you won't have this issue. And we really hope that some day YouTube's system will be compatible with Google's. But in the meantime, you can follow these steps:
Log into https://www.tubebuddy.com/account (this should be done by the person who "owns" the YouTube channel)
Click "Settings" under the channel that you want to grant someone access to. (Screenshot: https://www.screencast.com/t/vq4nWd6C)
Scroll to the bottom of the page, click the "Reveal URL" link and copy the full URL in the textbox. (Screenshot: https://www.screencast.com/t/JqdK3NO2s )
Email that URL to the person you want to give TubeBuddy access to. After they have the TubeBuddy extension installed, they can paste that URL into their browser and that will 'authenticate' them and allow them to use TubeBuddy features on your YouTube channel (assuming they are logged into YouTube.com with your channel)