Would Ebonia be free of organized crime thanks to its cashless society? Or could the Mafia or the Mob find another way to operate clandestinely?
No
Of course not. Where there's greed, there's crime. All a cashless society would do is change how the Mob operates — and not even that to a great degree. What are some of the many things the Mob is famous for?
The shakedown
"You need to pay $50/week to the following Venmo account to be protected against (*ahem*) difficulties."
Shakedowns are successful not because the financial transfer can't be easily traced, but because the victim is terrified to testify against the Mob. Any criminal activity that depends on similar terror tactics will prove successful using perfectly legitimate and traceable money transfers.
What would the Venmo line item description be? "Security consultation."
The Numbers Game
"Buy a chance to win the Lotto! Only $5/ticket! Legal? Well, not strictly... but see the Johnsons over there? They were last week's winners and they're on their way out to a fancy dinner!"
Illegal gambling has existed since monkeys learned how to carve bones into rough cubes and throw them. It's not the money transfers that are illegal, it's the reason for the transfer. Numbers games thrives because (among other reasons) if one person ratted out the Mob, everybody got into trouble. It's the worst form of "there's safety in numbers."
What would the Venmo line item description be? "Costillano's Investment Agency"
The Teamsters Model
"If you want to build your building, you need to deal with the Union! No, we're not here to talk about Jimmy Hoffa!"
The Mob learned a long time ago that political control begets financial control. Unions (and any other group with leveragable influence or an organized pension plan) could be infiltrated to ensure contractors paid just a little bit more for necessary services — and the Mob simply skimmed off the top took their consulting fee. Or they utilized the same organization's pension plans for reasonable investments. So long as the members of these organizations were basically happy, nobody stood up to the abuse... and keeping people happy is easy when the stereotypical "join the Union or else!" attitude keeps everyone in line.
What would the Venmo line item description be? "Teamster's #2072 Trucking Services"
Conclusion
Ignoring your title question completely, of course organized crime can exist in a cashless society. It's made easier with things like good cryptocurrency, designed to provide anonymity, and it's certainly made easier if there are countries where the money can be shipped off to without the hope of financial extradition. But organized crime can be reasonably said to have always existed and will always exist.
However, I wonder if you're thinking too much about a perfect solution. If you read a brief history of illegal transactions on the Dark Web you quickly realize that in many instances, it's the the people who didn't get sent to jail who move on to the next idea with new ways of avoiding capture. AKA, organized crime. If you think about it, mobster Al Capone, who everyone (and their brother and their dog) knew was a mobster, was so good at what he did that the way they finally incarcerated him was for tax evasion and not any of his criminal activities.
My point? Organized crime vs. law enforcement is the same back-and-forth escalation-driven drama as bullets and bulletproof vests.
And it's worth remembering that black market activities are still going on within the Dark Web in a wholly cashless "society" right now.
Illegal supply-and-Demand doesn't go away simply because paper money is replaced by numbers on a hard drive. Erasing value in a successful cashless society isn't as easy as erasing that hard drive. And there's a LOT of people (not just the Mob) who want some level of anonymity in their transactions, which leads me to believe a cashless society will never be 100% transparent — somebody will always be working toward the next great innovation in digital anonymity. Personally, I find a wholly transparent financial system unbelievable.
BTW, This question can be legitimately closed for Needs More Focus because the question in your title is very different from the question in your post. They intersect, but not enough to justify they're the same question. A black market does not mean organized crime must exist any more than organized crime means there's always a black market. You don't even mention a black market in your post