I'm considering soft resetting for them all since I've completed the post game. I'd assume they wouldn't be because the gen 3 starters aren't and they're the main starters of the game, but I'd like some confirmation. Thanks c:
Nope, they're definitely not locked. Professor Birch gave me a Shiny Chikorita in my AS yesterday after 1534 SRs! ^-^
Here's the live video confirming it:
And here's a full summary picture of my Chikorita: http://gyazo.com/38c7403978c8ccb69bc25fa8fef7269e
There are no evidence of that, as I know only some Legendaries or particular Event Pokemon are shiny locked.
Normal Pokemon obtained in game could be shiny, so you can try softresetting. I don't see any difference with the normal 3rd gen starter.