So in theory, the best Eevee spawn would be the individual who lives in Wild Zone 19. This is because it's the only one who shows up consistently regardless of the environmental conditions, whereas all the other Eevees either only show up during sunny daytime weather or only have a very low chance of spawning. This Eevee has a 5% chance of being Alpha, so also consider whether that's desirable or not.
Here's the problem: benches and fast travel points reroll the Pokemon in a 50m radius around you, and unfortunately, this Eevee's listed spawn point appears to be just out of range of any benches. From experience, I believe that you can actually see the Eevee itself in a few different spots among the bushes, but as it's not clear if there are various points for the actual spawner nor if they'd all be within range, I'm going to warn that the bench method may not work, or at least may not be as effective for this one. Of the people I've seen with shiny Eevees, none of them used a bench for the Zone 19 Eevee.
So you have 3 options for shiny hunting Eevee that I can think of:
- Repeatedly fast travel to Wild Zone 19. This is the most efficient method but it takes effort to do. May be possible to automate with third party tools that let you schedule a series of inputs.
- Use the bench by the northwest corner of Wild Zone 19, running several steps straight east up to the Wild Zone's edge every time to ensure that you trigger the spawn. Honestly, I can't see any reason to do this one over the previous one, but that's how you could use a bench for the Zone 19 Eevee...
- Use the normal bench method near an Eevee who only spawns during sunny daytime weather, accepting that you'll be reducing your chances by more than half because nights and non-sunny days don't count. This is the least efficient method but it requires the least effort and can be done AFK with a turbo controller. When you check for your shiny, make sure it's daytime with sunny weather (there should be a ☀ icon in the top right corner of your map).
Recommended locations include the south bench in Vert Sector 9 and the rooftop bench on the far western corner of Magenta Sector 9.
This analysis is based off of this map, which shows the 50m range and spawn points, and Serebii's map of Lumiose City, which gives more details about the conditions for each spawn.
