Eevee and its evolutions lean more toward (for the most part) carnivorous lifestyles, actually. While, like most pokémon, they can eat nuts, berries, grains, root vegetables, and so forth, much of their diet should be meat and protein of some kind.

That having been said, the answer to your second question depends. In captivity, the eevee family can actually be fed the exact same diet, regardless of what evolution each individual takes, with no real consequences so long as their trainer keeps in mind that they should be fed a balance of foods.

In the wild however, absolutely, it depends on the evolution.

Eevee tend to have the most balanced diet (as one would expect) and will eat whatever it can find in its immediate environment.

Flareon often stick to slower animals and pokémon, and its favored food sources tend to be small mammals. It will also display the unusual behavior of roasting their food with the intention of cooking it before eating. (Trainers should be aware that tamed flareon also display this behavior and should only be fed in fireproof environments.)

Vaporeon, being aquatic pokémon, will eat mostly seafood, including fish, amphibians, turtles, jellyfish, and aquatic mammals (such as seals). It tends to avoid shellfish, likely due to the difficulty in getting them open, but it will also hunt for larger meat sources (such as walruses, dolphins, and even sharks) in packs.

Jolteon, being faster and gifted with electrical attacks, often hunt for birds and forage for eggs. It’s also the one eeveelution that prefers pokémon meat over animal meat, as bird pokémon tend to be larger targets than their animal “counterparts.”

Espeon and Leafeon both tend to eat more of a herbivorous diet than their cousins. This does not mean that they are strictly herbivorous; simply that they forage more often than the others. Espeon tends to hunt for insects and other animals with lesser cognitive functions thanks to its psychic abilities. Leafeon, meanwhile, avoids insects (due to its plant biology) and consumes largely rodents, rabbits, frogs, and anything else it can find in its forest habitat. (Leafeon also photosynthesizes whenever a food source isn’t readily available.)

Umbreon, meanwhile, is much more of a carnivore than its cousins in that its diet consists almost entirely of meat. As a nocturnal, forest-dwelling pokémon in nature, it will often hunt for small, ground-dwelling pokémon and animals, particularly diurnal ones. It has no qualms raiding the nests of sleeping prey, in other words.

Glaceon will often consume whatever it can find in its colder, more arctic environment—usually seafood (for polar glaceon) or rabbits (for mountainous). Glaceon may also share territory with vaporeon, sometimes even hunting alongside them to take down larger prey (such as seals and even walruses).

Finally, Sylveon, like eevee, will eat whatever it can find in its environment but typically chooses prey it can lure into dark corners of its habitat and then devour.