It might initially seem ridiculous that Amber Mendez was able to smuggle the tape containing the EXTREMELY incriminating evidence on the government into the game show undetected; sure, she tells Ben Richards it's "none of your business" where she hid the footage, but c'mon, she was caught red-handed rifling through the filing-cabinet. Then again, the "crimes" they make up against her (which include having sex with 2, and sometimes 3 different men a year) indicate that this dystopian future society is rather sexually repressive; hence, its most loyal enforcers would be tend to be uncomfortable with searching through any woman's "special" hiding places.
Alternately, while the enforcers could bring themselves to strip-search her, they didn't have the time; she wasn't discovered rooting through the records until the night the show was on, and they were pressed for time to get her into costume to bring her onto the show.
Technically, the Really Gets Around smear wasn't indicated to be a crime, just a little extra dirt to go with her supposedly being the Butcher of Bakersfield's accomplice: "she's not only a criminal, but a slut too!" They were rather short of time to search Amber and Killian was looking to drive his ratings even higher by any means he could, including bringing on a mystery contestant. There's also the possibility that they didn't realize she'd actually stolen anything; the edited video was still in the cabinet and the raw footage she took was located behind it. While they did obviously notice where she was looking, her arrest was specifically for being in an off-limits archive, not the theft of the video.
Another possibility: they knew Amber Menendez had the recorded chip on her somewhere, but didn't care; as Richards' fellow contestants noted, their resistance movement actually had a base out in the game zone, but had never found the up-link satellite dish that was also out there because "No one ever comes out here." The people in the studio didn't know any of the contestants had friends out there and probably just figured that the recording (which they no longer needed for anything) would be fried, mangled, or just plain lost along with her corpse when she inevitably met some gruesome demise.
Where did the Resistance get their hands on the shots of the true fate of last season's winners? The scene where Fireball taunts Richards about them implies that it's not being filmed, or at least, it's nothing the audience is able to see. Did they have a guy with a camera run over and get a few shots after Richards supposedly told them about it?
Quite possibly they did. The corpses were apparently left in a location far enough from any of the TV cameras for Fireball to openly confirm their identities without fear of giving away the truth to Killian's viewers; provided they didn't run into a crew collecting the Stalker's own burnt remains, the Resistance could slip a photographer into the body-dump room without being seen.
Amber changes her mind about Richards when she sees two people on the news supposedly in the hospital in critical condition after being attacked by him, the same ones who almost let him get away; now, if a person portrayed as getting killed in fake footage after failing the state typically gets "removed" from reality as well, what are the odds that this totalitarian government allowed either of them to live?
Along the same lines, while he didn't die in the digitally faked footage showing him killing Ben and Amber, the state would surely deem Captain Freedom a failure and possibly a traitor too, and wouldn't want him around to testify to anyone that the whole game was rigged and his amazing "victory" was a total fraud...
On that point, Killian has Richards' and Amber's faces digitally mapped onto a pair of stunt doubles (he explicitly says that they are stunt doubles), who are then killed in order to make the world think that Richards and Amber are dead.
From the novel when Richards is going through the lengthy audition process, he finds the time to make a quick phone call to his wife; near the end of the call, she remarks that there's someone at the door. She and their daughter are killed before Richards even gets on The Running Man show... could they have been her killers?