No, you cannot currently catch the shiny version of the Mythical Grass-type Pokemon Shaymin in Pokemon GO.
Pokemon GO developer Niantic has not added the shiny version of either Land Forme or Sky Forme Shaymin to the game just yet.
Despite this, now is still the best time ever to catch Shaymin. This is thanks to the new Grass and Gratitude Special Research Story, which gives you a guaranteed Shaymin encounter.
Also, even if you cannot get shiny Shaymin, there are hundreds of shiny ‘Mon to catch. Find out the exact odds of encountering shinies in Pokemon GO!

Is There a Shiny Shaymin Release Date in Pokemon GO?
Niantic has not confirmed whether the shiny version of Shaymin will come to Pokemon GO yet. Therefore, don’t expect to see it any time soon.
In fact, only a few Mythical Pokemon have gotten shiny versions so far, and that is usually through a ticketed Special Research Story. Therefore, it will probably be a couple of years before we see shiny Shaymin!
For instance, players can get a guaranteed shiny Jirachi encounter from the Masterwork Research: Wish Granted.
What Color is Shiny Shaymin in Pokemon GO?
The shiny versions of both Land Forme and Sky Forme Shaymin retain the white primary color. However, they have a turquoise secondary color instead of bright green.
Also, the pink flower on Land Forme Shaymin and the red cape of Sky Forme Shaymin remain the same for the shiny versions.
Even if you cannot catch the shiny versions of Shaymin yet, there is a shiny Mythical Pokemon that is easier to catch than the others. However, you can only catch shiny Meltan at specific times.

Also, check out the list of every Mythical Pokemon in Pokemon GO. There are plenty for you to add to your Pokedex!