Because the Masuda Method doesn't exist in Generation III, the easiest way to obtain a single shiny Eevee is to save in front of the one in Celadon Mansion and soft reset until it's shiny. If you want multiple, your only option is to breed the gift Eevee and hatch as many Eggs as you can handle.
Both of these methods have shiny odds of 1/8192. You can collect Eggs faster by using two Eevee as breeding parents, and faster still if one of those Eevee is traded from a different game.
