There are 153 Pokémon available in Let's Go: the first 151 Pokémon of Gen 1, Mew, Meltan, and Melmetal. Because Let's Go can communicate with Pokémon HOME, there are effectively two separate Pokédexes that the player can complete for Let's Go Eevee:
Any Pokémon captured in Let's Go and transferred to Pokémon HOME will additionally count toward the HOME National Pokédex. The HOME National Pokédex will not show Alolan forms unless they are specifically transferred to HOME, although they do not need to be transferred from Let's Go in particular.
Let's Go Pikachu and Let's Go Eevee can both transfer Pokémon to Pokémon HOME, but can only receive Pokémon from HOME if the Pokémon both a) originated from a Let's Go game and b) has not been sent to any other HOME-compatible game before.
Pokémon caught in Pokémon GO can be transferred into Let's Go via the GO Park in Fuchsia City. The GO Park is the only way a Pokémon from Pokémon GO can be moved into a copy of a Let's Go game. All 153 Pokémon in the Let's Go Pokédex, except for Mew, can be caught in Pokémon GO and transferred via the GO Park. All of the Alolan forms available in Let's Go can also be caught in Pokémon GO.
As of this writing, Mew can only be obtained from the Poké Ball Plus accessory. It can be moved around between copies of Let's Go and sent to Pokémon HOME, but once it is transferred to any other HOME-compatible game, it cannot be returned to a copy of Let's Go.
In order to complete the Pokédex in Let's Go Eevee, you will therefore need access to either a copy of Let's Go Pikachu or Pokémon GO.
The following version-exclusive Pokémon have to be traded from another game:
Four Pokémon evolve by trading: Machoke, Graveler, Kadabra, and Haunter. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.