Legendary pokemon have the same chance to be shiny as wild pokemon, which is 1/8192. However, the difference is there's only one in each game, you don't run into different ones in the wild like you do with, say, Bidoof.
If you save before encountering the legendary, then you can keep resetting your game every time, eventually it will be a shiny, but only after thousands of times, which would take forever.
Note: for roaming pokemon the shiny status is fixed from when you first encounter it, it won't magically turn shiny after a few thousand encounters in the wild.
Also, if you get rare legendaries from events they are often shiny.