Currently none of translation team is allowed to activate this option, but the project owner. In fact, it is determined by individual language translation team whether the translation is required review or not. Furthermore, many of larger/matured Fedora language teams have their own policy on this and implemented manual review process already. It will be very powerful and useful feature for translation team if each team coordinator can turn it on/off depending on their context. Besides, I believe that generally project owners do not care about translation review really. Please consider to allow 'coordinator' to turn on/off 'Require translation review' checkbox.
As of Zanata 3.7, review process are enabled by default in the editor. You will be a 'reviewer' role to see the approved/reject option. See https://bugzilla.redhat.com/show_bug.cgi?id=1211134. We've remove the review option until https://bugzilla.redhat.com/show_bug.cgi?id=1211137 is implemented.
Migrated; check JIRA for bug status: http://zanata.atlassian.net/browse/ZNTA-49