My current theory is that Vodka has an eidetic memory. He isn't exactly a secretary or personal assistant, rather than a compendium of information that the BO can use and call upon. He was placed with the #1 assassin to keep tabs on all the people that Gin has killed in case a problem arises with killing said person and the BO needs to deal with it. If Gin cannot remember the conditions and the reasons and the circumstances of the person's death that could cause problems. He may not see a use for remembering but I'm sure Anokata does.
Consider this... Gin has supposedly killed so many people, and likely Vodka was with him the entire time. We know that a decent amount of time has passed and Gin has killed several people since killing Kudo, and yet Vodka was able to remember the exact moment, what happened, what time of day, and the method of killing Kudo.
Likely it extends past this. It is possible that he has other knowledge as well that would be needed by the BO. He was given a gun, training and placed with the #1 assassin. And the mere fact that we know that Vodka is not always with Gin means that Vodka is sometimes elsewhere on assignment, possibly by Gin's or Anokata's orders. He doesn't seem to have much common sense, but he appears to have an eidetic memory. I'm not sure what else he knows, but it can be assumed that it is more than the average person landing him in his current position.
Further to support this is that Gin does the killing, Vodka does the deals. He did the exchange at the amusement park and he was the person talking with Itakura about the program. He doesn't keep a log of anything it seems and yet he remembers the details and the conditions of all the deals. In this instance, Gin is actually aiding Vodka who doesn't know how to watch his back (like at the amusement park and at the drop off location for the program).
The only problem with Vodka is that he doesn't have as much common sense. But as Chek said, we know he has a good shot. I would say that he was the getaway diver, but that wouldn't make sense since he is always with Gin when Gin kills.
In fact, if he was simply proficient at bartering and making deals that could very well be the reason he gained a code name and was placed with Gin. Gin gets assignments to kill, and Vodka gets assignments to make deals. If a deal fails, it is likely Gin's job to finish the person off (and to make sure that Vodka doesn't expose them). No one else seems to make extortion deals or contact people outside of the Org for things other than Vodka. Vermouth has only been shown killing and assuming the identity of people, Gin has only been shown killing, the scientists only do science, Anokata remains in the shadows and dictates orders, the snipers snipe, so who extorts and makes deals for the benefit of the BO? Vodka. Would make sense.