If I have read the technical reports correctly, nicotine has not (at this time) been directly linked as a cancer causing agent. Tobacco specfic nitrosamines are supposedly present in trace quantities according to a number of sources, but at a level several orders of magnitude less than a regular cigarette. Nicotine is not without hazards, and it is possible to overdose. However, users would typically experience unpleasant symptoms long before this became an issue. Headaches, nausea, and increased blood pressure are early warning signs. This is usually sufficient to cause someone to decrease or cease their nicotine intake before a hazardous overdose would occur.
I'm no doctor, but I would think that unless you chew the same piece of gum for days or even a week, you would not be able to "trap" enough carcinogenic material in there to cause any concern.
You may want to consider that
vaping while chewing gum poses a higher risk of choking than anything else. While inhaling the vapor, you could accidentally draw your gum into your airway causing an obstruction that would prove difficult to remove if alone.