You're looking for an SSO plugin or the Session Sharing plugin.
Magento SSO Login
Has anyone ever developed a plugin to allow SSO with a Magento store?
We'd like to be able to have people use their existing login in our Magento store to login to NodeBB, but we are not too excited to develop it ourselves. Just curious if someone has implemented this before or would be interested in developing one.
Can't say they have. But if you followed the WordPress SSO I can't see it being too difficult. I miss magento. The clients that wanted a full store. But not pay more than £20 for it. Those that didn't appreciate how overly complicated magento can be.
@a_5mith Thanks for the suggestion! I'll have to see if I can use the WordPress SSO to help get started on a Magento one then.