For instance, while making a team, I tried to use a shiny Galarian Articuno. The teambuilder told me that Galarian Articuno was not able to be shiny, as it was only from an event. Are there any other Pokemon that cannot be shiny?
Pokemon Showdown only lets you use Pokemon you could legally obtain in a main series game, so Pokemon can't have moves they can't otherwise learn, can't have abilities they can't otherwise, and etc. This extends to all aspects of team building, so any Pokemon that are hard coded to never be shiny, or Pokemon that have no obtainable shiny without hacking cannot be used in Showdown. The full list of shiny locked Pokemon can be found here
Hope this helps!