There are 151 Pokémon in Yellow's Pokédex. Only 150 are needed to get the certificate from the Game Freak employee in Celadon City. The game is programmed to accept any 150, meaning if you use the Mew glitch to get Mew, you can skip another Pokémon. That said, this page focuses on getting the 150 Pokémon that aren't Mew.
In order to complete the Pokédex in Pokémon Yellow with only Gen 1 games you will need access to at minimum a copy of Pokémon Red and a copy of Pokémon Japanese Green/International Blue.
Access to Japanese Blue is not sufficient, as Ekans, Arbok, and Electabuzz are only available in Red, while Magmar is only available in Japanese Green/International Blue.
The following version-exclusive Pokémon have to be traded from another game:
Several Pokémon are mutually exclusive, meaning that if you get one species, you cannot get at least one other species. The list of mutually exclusive species is below.
If you have access to Pokémon Stadium, you can get all of these by defeating the Elite Four and Rival.
Otherwise, you will need access to two games anyway to finish the Pokédex. So, you can have each game choose a different Eeveelution, and then trade around to get all the Pokémon registered. Make sure that you trade each Fossil at each evolutionary stage, since you can't un-evolve Pokémon.
Four Pokémon evolve by trading: Machoke, Graveler, Kadabra, and Haunter. You can get Machamp via in-game trade, meaning you will only need to trade for Golem, Haunter, and Alakazam. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.
You only get one Pikachu or Raichu in the game, and it is the Pikachu you start with. It will reject the Thunderstone, so you cannot get a Raichu without trading. You can either trade for a Pikachu or Raichu from another version, or trade your starter Pikachu to another version, evolve it there, and then trade it back. Note that if you do this, Pikachu will no longer follow you.