There are three ways to get this one, two solo (Main Game) and one co-op (Main Game or Sandbox.
1) Co-op. Either you or your partner drink beer/whiskey/vodka near one or more zombies until it triggers vomiting. Try the bars in the Atlantica or Americana Casinos, or the challenge in the Atlantica Casino in Sandbox mode requiring you to drink so many alcoholic drinks (I forget the name). Make sure neither of you are wearing the full Sports Fan Outfit OR have the magazine for Drinking otherwise you won't get sick from drinking too much.
2) Main Game 1 -Drink 3 alcoholic drinks in a row to start puking near one or more zombies. Only do this early on if you are NOT going for the Purewal Memorial Cup achievement (Not consuming any meat/dairy/alcoholic items until after 9am on Day 4), or you can make a separate save and do it, then reload your main save.
3) Main game 2 - IF you're focusing on Purewal Memorial Cup before going for this one, I found a sorta harder yet less time-consuming way to get it than drinking any alcoholic drinks to excess (meaning with this method, you won't suddenly stop during a fight and puke). During THE FACTS when you exit out into Fortune Park and see Harvesters being set up, go for the nearest Harvester. One of the two/three guards will have the sonic blast gun and if he hits you with it, you drop whatever you're holding at that moment and you'll start puking. Hopefully that happens to you near a zombie and when you move away, the zombie will go after you and slip on the puke puddle. That's how I got it.