Usually cheats do disable all trophies, but for some reason it doesn't block them in this game. You can use cheats and trophies will still unlock.
I was able to confirm this by using the cheat to spawn a Rhino tank. I then triggered the vigilante missions and was able to unlock the trophy for passing the 12 vigilante missions, getting a 6 star wanted level and causing a large amount of damage to the city.
Others have also tested this and have been able to confirm that using cheats does not prevent you from being able to get all of the trophies in the game.
Looks like an already easy platinum has just gotten a whole lot easier!
For a full list of cheats that can be used in Grand Theft Auto Vice City can be found here.