Gary/ your rival's Eevee doesn't evolve based on your Pikachu's happiness.
The rival will eventually evolve his Eevee (and thus affect which are the other members of his party) depending on the player's actions early in the game:
- If the player beats the rival at both the Oak Pokémon Research Laboratory and the early optional battle at Route 22, Eevee evolves into Jolteon.
- If the player loses one of these two battles or passes up the Route 22 battle, Eevee evolves into Flareon.
- If the player loses at Oak's Lab, and loses or passes up the Route 22 battle, Eevee evolves into Vaporeon.
I guess your Pikachu's happiness is linked, as winning battles will make it more happy, but that in itself doesn't determine your rival's Eeveelution.
Pokemon Yellow - Bulbapedia