Does the NEC allow electrical wiring to run next to a ventilation duct?
There are a few things I can find in the NEC that may apply here. NEC Article 300.8 states that no raceway or cable tray can have elements of any other system besides electrical in it; no plumbing, no fuel gas, no ductwork. This implies that using the vent line itself, or any of its mounting hardware, to support the cable is a no-no; if the wire and the duct go in the same direction for a while, each system needs its own supports connected to the structural members, independent of any other support for any other subsystem in the building.

NEC 300.22 specifically covers wiring within spaces that handle exhaust or environmental air. Basically, an exhaust vent line cannot have any electrical wiring installed within it. Not nonmetallic, not MC, not rigid conduit, nothing. So, you can't drill a hole and fish the cable through your range hood vent. This is because the vent line carries exhaust gases and particulate matter that may be flammable (in the case of a range hood, that would mainly be grease particles suspended in the hot air from frying, and also uncombusted natural gas), and having a wire that could, in some imaginable case, cause a spark would be an extremely bad thing to have in a line with flammable gases or particulates.

As an aside, for environmental air (as in your HVAC system), the NEC just states that the cable must be plenum-rated in order to be legal in an open space used for air handling, and must be armored in some way when installed inside a metal air-handling duct.

Additionally, NEC 300.4 covers basic protection of nonmetallic wire against damage. These include things like minimum depth of installation in a wall, properly anchoring electrical wire to structural members etc etc, but also covers protecting the wire from sharp metal edges. Ductwork can have quite a lot of sharp metal edges; extra protection, such as bushings and rigid conduit, are recommended and/or required for any run of cable that passes close enough to ductwork etc for contact to be a concern.

Given that you follow those three articles' guidelines, I cannot find anything saying that Romex can't be in close proximity to ductwork. I would just avoid having the conduit actually contact the ductwork by securing it to the nearest structural member, and I would protect it with plastic or metal rigid conduit if contact with the metal duct is unavoidable at any point.