A small group can intimidate a large group by pinpoint assassination of those in power, their siblings, parents, friends, and (for all of those people) their children.
The rebels do not have to kidnap people to hold them hostage: they just need to be good enough (and heartless enough) that if the politician or general or cop refuses to comply with their private (not public) demands, people they care about start dying. In car accidents, of illness, of drug overdoses or accidental medication issues or heart attacks at 29. Whatever. If the politician talks, they are gone or discredited or framed for pedophilia with child porn found on their computers: And the rebels have in their pocket, developed for years by the same tactics, judges, prosecutors, police chiefs, Congressmen, Senators, and so on (or whatever the equivalent is, in your world).
Reportedly Al Capone once said, "99% of men will fold up the second you cut them. The other 1% work for me."
Your coup is silent, nobody knows it is happening. The rebels do not ask for much, but when they do, deliver: or your daughter in college is about to die of a drug overdose at a concert, just like her dorm roommate did three months ago, and nobody could believe it. But you believe it, because you got an instagram of that girl with the words "the prime of her life" scrawled on it, and you didn't know what that meant. Until that night she died. A little hacking (obtained willingly or by intimidation) and there is no record of the photo or any proof of their messages.
Your rebels can be smart enough to play a long game; take a small city first, councilman and a mayor, a city manager and local cops and judges that dare not disobey their infrequent requests to look the other way, lose some evidence, accidentally throw a case. All just as an exercise in building compliance.
By the time the rebel leader actually confronts the President; he's terrified for his life; it is about to look like he committed suicide in the Oval Office (or whatever your equivalent is).
But make sure your rebels are never brutal for no reason, and do ensure rewards for compliance. Whispered words will travel, they may be brutal and ruthless, but they keep their word. The rebel leader gives his word, the President will live in comfort and no harm will come to him. As long as he keeps his mouth shut, his pockets will be full. If he cannot keep his mouth shut, his head will soon be empty. He hands over the codes; everybody that was supposed to protect him is dead or under a rebel thumb.