I would really like to attempt this run however actual NES are hard to come by and emulators feel weird for me. I do own an NES Classic which has LoZ pre-installed on it, would this be allowed? It is not explicitly mentioned in the rules.

Hello

Yes I think it s allowed on the NES classic mini you can find it in the dropdown "Platform" when you submit your run

Yes! If you look in the left hand column (on PC) you'll see, below the game's cover, all the platforms that are allowed. The NES Classic is allowed and is perfectly fine :)