Short answer: No. Halo 3: ODST has never been released on any Nintendo platform. The game is a Microsoft/Xbox exclusive through and through.
The Halo franchise is owned by Microsoft and developed by Bungie (at the time) and later 343 Industries. As a flagship Xbox exclusive, it was never ported to competing consoles. The Wii's hardware architecture and Nintendo's licensing policies also made such a port technically and commercially unfeasible.
No Halo title has ever appeared on a Nintendo system—not on Wii, Wii U, Switch, or any handheld. The closest you'll get is Halo-inspired indie games on the eShop, but the real deal remains Xbox/PC only.
If you want to experience ODST's noir-infused campaign and Firefight mode, you'll need an Xbox console or a gaming PC.