It appears that only the site manager has the GEDCOM export option under Manage Tree. I am a site member of a family site and the only option I have under Manage Tree is view. Is this a permissions issue with the site?
It is true that only the Site Manager has access to the "Export GEDCOM" feature on the website.
What you can do as a site member is to either use the "Download" Zedcom feature if the tree was created via Family Tree Builder, or you else you need to contact the Site Manager and ask him to for the data