Is it possible for NPC trainers to have a shiny Pokémon?

If you mean a random shiny then no, NPC trainers always have predefined Pokemon.

The only case I've battled shinies was at the Trainer Tower in Pokemon FireRed/LeafGreen.

From Bulbapedia:

The Trainer Tower has three Trainers who use Shiny Pokémon: Burglar Jac has a Meowth, Twins Jen and Kira have an Espeon, and Fisherman Kaden has a Seaking.

In the Single Battle mode, the seventh Trainer has a Shiny Meowth, while in Double Battle mode, the first pair of Trainers have a Shiny Espeon. Finally, in the knockout mode, the fourth Trainer will send out a Shiny Seaking.

Here's also a video proof of shiny Espeon.

From Trivia here

There's some NPC Trainer that can be found with shiny pokemon.

1. The Minccino that appears in the "world of Pokémon" intro in Pokémon Black and White.
2. Rental Pokémon found in the Battle Factory in Pokémon Emerald.
3. Battle Frontier Pokémon used by the AI in Pokémon Platinum.
4. The Ralts that Wally catches in Pokémon Ruby, Sapphire, and Emerald.

There's also shiny pokemon that can be found not in the wild,

1. The Pokémon that the old man encounters in FireRed and LeafGreen.
2. The Poochyena/Zigzagoon that attacks Professor Birch in Pokémon Ruby, Sapphire, and Emerald.
3. The Purrloin that Bianca catches in Pokémon Black 2 and White 2.
4. The Fletchling that the rival, Calem or Serena, uses in the capture tutorial in Pokémon X and Y.

Other than that, what pinckerman Said is true and I'm also think that NPC trainers always have predefined Pokemon.

Excluding trainers that always have shinies, yes a trainer can have a shiny Pokémon if and only if the Pokémon that they have are randomized. Most trainers in the games have a predetermined set of Pokémon and this won't happen to them, but in certain cases, NPCs will have a random set of Pokémon.

This actually happened to me. I was going through the Battle Tower in Gen IV and my opponent had a shiny Girafarig, the only shiny I had seen in my 400+ hours of playing the game. Needless to say, it was a frustrating experience that I would not forget.