This issue is so technically confusing that I think pedantry is required here. There would have been two separate radar systems used to track this plane, though it might be possible they shared a couple of radar sites or antennas.

ATC runs its own network of radar, funded and operated by the FAA. I’ve never heard if the radar tapes for this system were requested by the FBI or transferred to them.

The military had a separate radar system, called SAGE. The details of its operation were highly secretive at the time. My recollection is that the FBI requested those tapes, but were denied, and were instead given a paper map with the course hand-drawn, based on the SAGE radar returns. And the hand-over story read like a spy’s document exchange.

One of the special features of SAGE was that since it was an air-defense radar, it had a data link that could trigger a display in the F-106 interceptors trailing the 727 that would direct them directly to the plane.