On a quick search. I have not found any results with the queries I used to force PMs to require moderation
Someone may have better info on this. You may need to consider making a request in feature and if need this more immediately marketplace and sponsor a plugin.
Can you share some info on your community idea with the need to have PMs require moderator approval?
If members do not need to be able to pm each other. But you still want members to be able to pm staff. There is a plugin that ensures users can always on staff(moderators)
With this plugin you could disable PMs.
Just was reading over the “Private Message Keyword Scanner”
Looks like is broken atm. It looks to be otherwise the closest option though doesn’t flag to require approval.
The built-in Watched words feature could work if there are keywords/phrases - but will affect public and private posts
Would need to look into this. But you might be able to use Automation plugin to auto flag pm for approval.
This plugin is now part of core so no installation of it is needed.
Edit: posted a question in the Automation plugin topic regarding your requirement.