Yes, you can catch the shiny version of Azelf in Pokemon GO, but only if you are lucky.
The odds of any Azelf you encounter being shiny is 5%, or 1 in 20. This is the standard shiny odds for Legendary Pokemon in Pokemon GO.
If you are searching for a shiny Azelf, then there are a couple of events you won’t want to miss. The two Pokemon GO Azelf Raid Hours are the perfect times to catch this shiny Legendary as it will be appearing as a Raid boss much more often.
However, remember that Azelf is a region-exclusive Pokemon. You will only be able to battle Azelf in Raids in The Americas or Greenland – it isn’t available in the rest of the world.

What Color is Shiny Azelf in Pokemon GO?
The shiny version of Azelf is a bright yellow color and has a blue head. This makes it appear very different to the standard version of Azelf which is gray-blue with a teal head.
Despite the stark differences, both versions of Azelf have yellow eyes and red diamonds on its tail and forehead.
In addition to shiny Azelf, you can catch the shiny versions of the other two Legendary Pokemon in the Lake Trio – Mesprit and Uxie:

There are dozens of Legendaries in the game, and most of them have shiny versions. Check the list of every Legendary Pokemon and Ultra Beast in Pokemon GO to see which ultra-rare shinies you’re missing.