Odds are there isnít 12V at the vent. You could get 12V there a couple of different ways. If there is a ceiling light close by you could tap into that. If the lights are all on one switch you would have to have the lights on to operate the fan, if the lights have individual switches it would be much easier.
Another option is to find the 12V supply for the stove vent and then fish over to the vent. There may be some constant hot 12V wires running in the ceiling on either side so your not out of luck, just may take a bit of work.
2012 Copper Canyon 273FWRET being towed by a 1994 Ford F350 CC,LB,Dually diesel.
Airlift 5000 bags, Prodigy brake control, 5 gauges on the pillar.Used to tow a '97 Jayco 323RKS.
Now an RVIA registered tech. Retired from Law enforcement in 2008 after 25+ yrs.