Officially, the Mongols destroyed the Assassins in Outremer in 1260, in both our timeline and in that of outremer, yet in our timeline, Assassins were employed by the Egyptians for some time after 1260. In the section on Homs, it is brought up that the inhabitants of Homs consider the Assassins not gone, but "occulted", hidden but still active.
You may wish to work with this suggestion as fact - either the Assassins did survive the Mongols, or someone has reconstructed them in imitation. Either would be effectively the same - and by the time of the game. 200 years after the Mongols crushed them, these Assassins may truly believe that they are the survivors of the Mongol attacks, whether or not this is a fact.
Unlike the old Order of Assassins, who carved a demesne out of first Antioch and then Homs, these new Assassins would be quieter, keeping to the shadows, and hidden. I suggest hiding them in plain sight. Good places to hide would be as a feudal lord and entourage, as religious types, or even as a company much like that of the player characters'.
Contact with the Assassins would then have to be either initiated by the Assassins themselves or through a double blind contact through word of mouth. I think initiation by the Assassins would be more characteristic of them. The Assassins did not just work for hire. They worked toward their own ends while allowing others to finance the means to those ends.
One thing the historical Assassins did was always play one side against another, never letting one side get too strong. In this sense they could be working for the good of all, despite their unsavory means. Is it a good thing to kill one person to stop a war? They may think so. Alternatively, They may be doing this to keep everyone else down until they can take over everything themselves, in an entirely selfish power grab. Either viewpoint works.
As for means, the historical Assassins worked by suborning trusted people in the inner circles of whoever they were targeting - and not by money! They created fanatics out of what would appear on the surface to be solid, respectable citizens. How did they do this? No one knows. The story that they were given hashish and "taken to paradise" appears to be a fabrication, perhaps even one planted by the Assassins themselves. Perhaps it was magic, or the promise of power, or even a return for a great service performed by the Assassins. All of these are possible, and maybe all of them are used. Certainly, the Assassins would never give out their secret.