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This can easily be obtained on survival as Tanks are frequent and every match normally starts with Bile close by. Simply throw your vial of puke at a Tank at any point in the game and this achievement will pop.

Non-survival method:

I suggest starting on the campaign "Dead Center" and selectively killing the hazmat suit zombies. These zombies will drop puke vials often which you should collect and save until you encounter a Tank during the campaign. There is also an achievement tied into collecting puke from the hazmat zombies.