It's not clear what the feds knew but there's absolutely no indication that the govt knew that Syndrome was killing supers, nor that they wouldn't have immediately tried to stop him if they'd found out.
As to the question of why the National Supers Agency aren't doing something about it, the people tasked with hiding the supers (notably Rick Dicker) do appear to have knowledge of their secret identities but only seems to involve itself in their affairs when they accidentally reveal their powers, and then only to help the supers to remain hidden rather than investigating what they're up to.
Beyond that, the detection of mysterious superhero disappearances appears to be a simple police matter, one dealt with by "local authorities".
There's also a few other things to take into account:
The implication is that the "Supers" department has been heavily de-funded since the supers were forced underground. Rick Dicker complains about the extravagance of having to relocate the family again.
The disappearances have happened over an extended period of time (certainly years and potentially decades), making any sort of pattern hard to spot.
The missing superheroes are likely to be from all over the US, meaning that there's less chance for the connection to be made by local police.
Mirage actively encourages the supers to lie to their loved ones about their movements. The police are far less likely to assume foul play if they find that the disappeared person has been leading a double life.
The supers lack any sort of centralised communication among each other. The winnowing of their numbers has evidently gone unnoticed.