The reason why you shouldn't "smoke" with dentures is because smoking cigarettes can leave a film of tar on any oral dentures and stain them. This is the same with your natural teeth and gums, smoking can stain them both. Vaping doesn't stain anything so you can vape with dentures on with no problem in my opinion. I was a dental assistant, so I know a little bit about these sorts of things.
You should still clean your dentures regularly because of any food or drink that can discolor your dentures though, so just keep that in mind.
The thing about "dry socket" is completely different. After a tooth extraction, you basically have a sensitive hole in your mouth that needs to heal. Sucking on anything, whether it be from a straw, cigarette or an e-cig creates a vacuum in your mouth and can irritate the part in your mouth that is trying to heal. It'll basically suck out the scab in your mouth and damage it. If you do, the blood that was in the hole that's trying to heal might be removed and you'll end up with "dry socket" which is a hole with no blood for it to heal. What you have is basically a wound with bone and nerves right where the blood is supposed to be so even things like breathing in air could be very painful.
Don't vape after a tooth extraction. And IF you REALLY need to vape, take an EXTREMELY slow drag. Don't quote me on that since there's a chance that dry socket could still occur. I'm just putting it out there. Also avoid overly hot and cold foods/drinks.