The first article I found on Google says:>Only primates, such as monkeys and apes, and horses have lots of sweat glands that allow them to sweat as much as humans do.>Some mammals that do have a few sweat glands produce small amounts of sweat through their paws.
This brings me to a question I hadn't thought about before: what is the length of the fur on an Eeveelution? Besides Vaporeon and Glaceon, the rest have tufts of fur and/or large collars of fur that must be longer in length than their body's coat. Is the main fur very short, like a horse's hair? If so, Umbreon sweating would look something like this, but purple: https://springhillequine.com/wp-content/uploads/2020/08/Sweating-horse.jpg
It would make sense for the fur length on Eevees to be closer to that of a cat/dog/fox/rabbit, but they're drawn and modeled as slimmed down anime animals, so it's up to our imagination. If they are short or medium haired like one of the animals above, Umbreon might look like this: https://image.shutterstock.com/z/stock-photo-little-monkey-sweating-on-a-hot-day-31870856.jpg
This leaves me with more questions, too; does Vaporeon have fur or slippery seal-like skin? Are Jolteon's spikes made of fur or quills like a porcupine? Are Leafeon's ear and tail leaves real plants, or are they animal cells mimicking the appearance of a plant? Are Umbreon's rings lit up by bioluminescent skin underneath or does the fur itself glow with dark type energy? None of it matters, but it's fun to try to apply real world animal logic to Pokemon.