The way I see it the conspiracy would have to pre-date any space program.
Prior to and up to WWII planes and ships used celestial navigation. The math and data, for the sake of argument, revealed to people not in the conspiracy involved geometry and astronomy based on a round Earth. Astronomers and some cartographers would need to be involved.
When building long suspensions bridges it is taught the curvature of the earth needs to be taken into account. So at least any engineers designing the bridge and some construction workers need to be involved. Plans would need to be presented that take the curvature into account, then other plans used for the actual bridge. A special crew would be needed to construct the parts that were the hidden plan.
Surveyors would at the very least need to be carefully watched and monitored. It could be very easy for them to start noticing a inconsistencies. Simply because a FE map and RE map would not mesh well together. They would have to fudge in some areas. Maybe only allowing their surveyors in certain areas where this inconsistencies would be more noticeable would work.
At least some pilots and ship captains and crews for certain routes. Again for the map being inconsistent with actuality. If this is not true you would have pilots and captains wondering why they are burning more fuel, landmarks not being seen, taking longer then planned.
Meteorologist may need to be involved. I am not talking about the ones you see on TV, I am talking about the ones studying the atmosphere.
Many people involved in the Earth Sciences. Geophysics, geostatistics, volcanology, paleomagnetism, seismology, petrology and hydrogeology is what I can think of. All could lead to the discovery or have discovered the Earth is flat. Either they need to be monitored or part of the conspiracy.
Of course there would need to be people doing the grunt work keeping things under wraps.
Then the leadership of the organization/people at the top would need to pass the torch. I guess to their children? Maybe they induct people after a thorough vetting process.
I have no clue how many people would need to be involved. I find it hard to imagine it is under a 1,000 as I have seen some people claim. I do think it would have needed to start before the space program. I know some FE'ers do not like to admit it but there is a whole lot of data obtained and scientist were agreeing the Earth was round prior to 1961. So if they were selling a fake RE theory and discounting a FE as being implausible to impossible it stands to reason the conspiracy may have started before they claimed someone made it to space and orbited our planet.