Someone actually did complete a run of Pokemon Yellow with only Pikachu, but he used death fodders because Pikachu is extremely frail.
Things I've learned from the run (if anyone is planning to make one)
1. Brock is extremely hard with only Quick Attack.
2. You basically need Seismic Toss in order to get past Rock/Ground types as well as Ghost Types (seismic toss can damage ghosts in RBY) , since using Submission means you'll have to carry healing items, which isn't good for a TAS.
3. Body Slam is used because of being stronger than Mega Punch, as well as having 100% accuracy. In a TAS, Body Slam could be used for the Paralysis chance instead of waiting for a 1/256 miss.
4. Self-Destruct is the easiest way to get past Geodude.
5. Mimic is needed for Giovanni and the E4 Fights. (Giovanni's Dugtrio knows Fissure, Bruno's Hitmonchan knows Ice Punch, which is needed for his two Onix and Lance's Dragonair knows Ice Beam.
And if anyone wants to see it:
http://www.twitch.tv/mountebank/c/2978092