Another option I found extremely helpful, was using the same location, Vault 21 to pickpocket the bubble gum like others. Except I did it a little different. I only had a sneak on 64, and even using the stealth boy, I still was getting caught, no matter if she was sitting or leaning against the wall. Heres what I did.
Get behind her and pick pocket the bubble gum. If successful press select to wait an hour. This will also "autosave" your progress. If you dont return to her inventory screen, meaning shes catches you, press "start" and reload the auto save. Rinse and repeat until the achievement pops.
I found this way much faster plus as a bonus you wont lose and successful pickpockets by only saving after every 2or3.