How do I make an entity give an effect to another entity after hitting it with a projectile

Make an unobtainable ranged item with the properties and appearance you want; then, make a procedure under 'When Projectile Hits Living Entity' that adds the poison effect to the target entity. (Make sure it's the target entity, not the source entity, which would poison whichever mob shot it.) Then, for your custom mob, in the AI settings, make it shoot the custom ranged item for its attack.