I work at a hospital which has a "no tobacco" policy. I've carefully read the hospital's policy of no tobacco products, and no where is there a hint of mention of e-cigarette use or alternative nicotine delivery devices. It only mentions cigarettes, tobacco, and smoke.
Now whether the powers that be consider nicotine itself as a direct product of tobacco, I beg to differ. Nicotine is present in patches and gum, and there is no mention of them in the hospital policy. I have no tobacco in my presence, and I am not creating smoke by vaping.
Having said that, I stealth vape at work. I keep my gear in my locker, and often go there on my breaks for a few puffs, discreetly. My locker is conveniently located at the far end of the room, and behind another row of lockers, so I can easily see who enters the room. I know who I can trust to vape in front of, and abstain if in the presence of people I don't trust.
It's sad to say that I have to go through this process, considering that I have chosen a safer alternative from smoking, but it is what it is.