You absolutely can! The chances of it appearing are pretty random.
You can get it a few seconds in or you have to wait a little while
until it appears.
But, in my honest opinion and from my experience, it is not really
worth waiting for a shiny. It might give you more tickets than the
others, but most shinies I caught there gave me as much as any
regular Ducklett.
I feel like the most valuable is the baby ducklett. It spawns
uncommonly, but often enough that you dont have to wait minutes for
it, and they give around 7-15 tickets.