Well... yes... and no...
What one country considers illegal another country may not. That's true today all over the world. Take the practice of female genital mutilation, for instance. Most western countries absolutely abhor that practice — to the point of some really large multi-national non-profits acting diligently to bring it to an end.
Is female genital mutilation legal in, for instance, the U.S. No. Is it generally abhorred in the U.S.? Yes. Is the U.S. on the verge of landing the marines in countries that practice it to stop the practice?
In a sense you've asked a question that's too broad. There are thousands, maybe even millions of practices that one country would consider illegal and yet are practiced in another. And they're allowed to do it because (a) no country actually has the moral right to force any other country to conform to its beliefs (oh, they try...) and (b) most countries either have something the other country wants (oil, cheap labor...) or is big enough to push back on an effort to stop it.
So, from one perspective. Absolutely. A bandit country more-or-less like you're describing can exist in the modern world.
You appear to be describing what we'll call... the Free State of Tortuga. At least the Tortuga of the 1600s made popular in Disney's Pirates of the Caribbean movies. Could such a lawless place exist?
Not really. Oh, we have our drug cartels in Columbia and our human trafficking out of Somalia and as much graft as you can imagine in Mexico... but you're describing an oxymoron... a place that's completely lawless and yet under the control of other nations. The problem is that those other nations require results and the bandit nation is in the business of giving them the proverbial vulgar hand gesture.
So, while there are plenty of examples of modern countries that do absolutely deplorable completely breathtakingly evil things (Nazi Germany's enactment of the Jewish Holocaust comes to mind as does the subjugation of women in too much of the Muslim world), the reality is...
You can't depend on a bandit country to do anything useful for you. The moment you've enforced enough order to get predictable results, you no longer have a bandit country.
People need to eat, get medical care, etc. Yes, they can, have, and do live without great amounts of those things, but the more they must fight to stay alive the less time there is for banditry — but the less they have to fight to remain alive, the less the general population wants banditry. Laws are imposed by the largest group with the mightiest weapons. And those nations might be interested in selling the peons a few. At which point you no longer have a bandit nation.
Finally, you appear to be describing, basically, chaos. Sounds like a place you want to visit on spring break, right? Yeah, if only to avoid liability, most nations would be telling their citizens that they visit that nation for a hedonistic vacation at their own risk — and it wouldn't take more than one or two high-profile murders or kidnappings before everyone who might want to go there... stops going there. Why is that important? Because if the only export from your island country, which needs imports of foods and materials to survive, is considered illegal everywhere else... Let's just say it's a lot easier to stop an island nation from exporting cocaine than it is a cartel in Columbia that has an entire continent worth of coastline and landmass to work from to sneak something out.
You've created a easily located, hard-to-defend, impossible-to-feed lawless nation and hope that it can realistically exist in the Real World.
But examples of different aspects of your lawlessness exist today all over the world. What you generally won't find is that the criminals are working out of an easy-to-find hard-to-defend island. Instead they're working out of large landmasses with population centers that allow them to hide behind the proverbial human shield.
Frankly, if such a place existed, the first world leader to vaporize it would get booku credibility. And the first world leader to get caught using the services of such a high-profile criminality would be lynched.
And maybe that's the problem: you've created a high-profile example of a service that's traditionally very, very, very low-profile... for a reason.