No. The skin doesn't work like that. Your liver and kidneys remove that stuff from your body, it doesn't come leaking out the skin even when you sweat. And if it does, it's in trace amounts and totally inconsequential. And I've never heard of a rash being a symptom of withdrawal.
What you're experiencing could be something like psoriasis or dermatitis, dermatitis can flare up at even the most minor irritations. The other possibility is you're allergic to something. I'd say it's been long enough vaping to not be anything in the juice, but allergies can come and go for no reason at all. And with allergies it really could be almost anything you're exposed to during a normal day. I'd recommend getting some hydrocoritisone cream from the pharmacy, it's cheap and effective against stuff like this. It's basically normal lotion with an anti inflammatory steroid in it to calm everything down. If it still persists, make an appointment with a doctor to have it checked out. They can give you something stronger to put on it or refer you to a specialist. You can also try some over the counter allergy pills which there are many varieties that all do pretty much the same thing.
Good luck.