Can I run an extension cord through my crawl space?
Would it be to code and pass a city inspection? I highly doubt it, particularly because it is through a vent and there are specific code requirements for running electrical cables in that scenario.

Will it work? Yes.  

I'd say if its for a short-term/temporary measure of light loads (a light) it will be OK, but if you plan on it being permanent or running heavy loads (an electric lawn mower), you should really have a new water tight outlet properly installed.