This Instructable,
http://www.instructables.com/id/An-Inexpensive-Smoke-Fog-Generator/, points out that the only part of the vape
unit you need is the heating
element, which can be as cheap as several for USD$8. Pump 3.7v through it while it is sitting in a pool of food-grade glycerine and you have your
smoke.
Now, the motorcycle exhaust
effect. Motorcycles have a big puff when the engine is started, and then short bursts thereafter. Probably what you'd be looking to do, with the above Instructable
unit, is have a small powerful fan hooked up to it. The heating
element would need to kick on the instant the "key" was inserted into the motorcycle, then when the engine is revved the fan kicks on for a big blast of
smoke. Thereafter the fan needs to operate in short bursts, to give the heating
element the opportunity to rebuild some
smoke supply.
You could have a
stagehand in the wings, or a lighting technician with some creative
DMX work, run the
system manually. Alternatively, a cheap USD$10
Arduino with a USD$10 2-Relay
unit can run both the heating
element and the fogger with control from the actor: insert key, fogger starts engine and puffs while running; remove key, fogger shuts off.