First of all, to kill someone with poison in Hitman 2 you have to analyze the behavior of the target - what they eat, what they drink, how you can get them to "take" poison. The second step requires you to obtain the poison.
Only poison marked as deadly will neutralize the target - poisons such as emetic rat poison or chloroform will not pass the test (chloroform will cause the target to fall asleep, but its subsequent killing is not treated as poisoning when completing the challenges; rat poison causes only indigestion - the target looks for the closest toilet or other place where he or she can vomit).
When you reach mastery level 7 in Whittleton Creek (Another Life), you can equip Agent 47 with deadly poison at the start of the game, instead of searching for it on the map (especially since you can't always find it - it doesn't occur in the first mission, for example). Attention - the poison is one-time option only. However, you can place another bottle on the map (smuggling).