"Is this a thing? A prisoner can admit to murder, get sentenced to 30 years and the apply for clemency?"
Actually, there's nothing all that odd about someone asking for clemency who did a crime. By its very nature, "clemency" is a request to be given reduced punishment for something that the court determined you did, and in a way, clemency is just an exaggerated form of parole, where the person that's convicted is being allowed to serve less time than the trial verdict says. The Arkansas Parole Board will be the ones to decide. In fact, some of the worst criminals sometimes are granted clemency here or there, typically after serving many years and then being terminally ill and about to die.
Yes, at times clemency is also used to free a person with a questionable conviction, but usually those situations are resolved by a new trial or a pardon rather than clemency.
IMO This isn't a big deal, really, as the only oddity in it is making such a request so soon after being convicted of being a part of a murder plot.
But it's no big deal for them to be making a decision, because they have no choice. When she asks, even if it's a ridiculous ask, someone has to make a decision. But I can't imagine it will get any real consideration. She helped kill someone, in a cold brutal crime, and has barely served any time at all. The board officials didn't get their jobs by being naive idiots.