All information comes from Bulbapedia. Game Freak never really reveals what Pokemon are based on, so most of the below is speculation.
Vaporeon's origins are uncertain, though it could be based on a cross between cetaceans, felines, and canids. Its feline and fish characteristics may mean that Vaporeon may be a visual pun on catfish. The fins and dorsal lining may be based off different types of tropical fish. Its ears resemble a mix between a fish's fin and a fennec fox's or caracal's ears. It also shares characteristics with mermaids. Vaporeon may also be inspired by the Telchines of Greek mythology, or the Merlion, a national symbol of Singapore. Vaporeon's resemblance to both carnivorans and cetaceans could be based on primitive whales, such as Kutchicetus.
Its physical anatomy seems to be primarily based on a Herding dog or possibly a fennec fox, with several traits of coyotes. It is also feline-like; its extremely high speed resembles that of a cheetah's. The ability to shoot its spiky fur may be based on porcupines. It may also be inspired by the raiju, a beast from Japanese mythology whose body is made of thunder, and may take the form of a cat.
It is most likely based on foxes, due to its body shape and orange coloration, but it also has traits of felines. The fluffy mane and tail resemble a Pomeranian dog, as well as a lion's mane.
Espeon appears to be based on legend of the nekomata, also known as bake-neko, which are mysterious, intelligent, fork-tailed goblin cats. It additionally resembles a carbuncle, a Latin-American mythological beast which had a jewel on its forehead. It also has traits of caracals, sphynx cats, lynxs, and fennec foxes.
Umbreon's origins are uncertain, though it could be based on a black cat or possibly a black fox, a color variation of the red fox. Its long ears also are similar to those of a hare; numerous East Asian legends concern a Moon rabbit. The rings on Umbreon's body look like Egyptian lunar markings and jewelry that have been associated with animal sculptures, such as cat statues of Bastet and statues of the jackal god, Anubis, from Ancient Egypt. They may also be meant to resemble solar eclipses.
Leafeon bears many similarities to cats and foxes, such as its behaviors like tree climbing, and its cry sounds similar to a cat's meow or a foxes chirping. Its coloration appears to be inspired by the markings found on animals like the Siamese Cat or the Red Fox. It also has traits of plants, such as the ability to utilize photosynthesis. The brown marks of its paws give the appearance that they were buried. In Japan, and around the world, there are stories about domesticated animals that upon entering the forest become wild and eventually become one with it. Leafeon could be an exaggerated example of this story.
Like many of the Eeveelutions, Glaceon shares characteristics of foxes, dogs and cats. It is more closely based on an Arctic fox, due to their habitat. However, its large ears are similar to a fennec fox.
Summing it up, most of them have a strong resemblance to foxes, cats, etc. or other animals with a similar body form.