Can I pause buddying an eeveelution.

No you can't , distance have to be continues

Just keep walking it and only evolve it at night time. You'll lose a few KMs for your Chansey but you'll be sure to get Umbreon.

You need 10 km and 2 candy. Get that candy, switch it, walk it a bit more in the evening to be sure, and evolve through buddy menu, should be fine. "Night time" is anytime the screen is dark, even if it's like 6 pm in winter and doesn't feel like night at all.

You need to do at least 10kms
You need to get at least 2 candies
You need to evolve either night or day depending on your choice.
You need to have them as your buddy when you evolve.

These can all be done at once, or all at different times with breaks in the middle. You can do the walk, and then unbuddy - but as long as you buddy up before the evolve it will work.
That is, all these things needs to be done - but they don't need to be done all at once.

You can pause it only if you've been walking it since buddy distances were decreased during the rock event. I had one that I'd walked about 8.5 KM at the time, so I immediately got a 2nd and 3rd candy when the rock event began. I swapped it out and didn't make it my buddy again until the event ended a week later. Then I walked that last 1.5 KM or so and evolved it, and successfully got an Umbreon.

But chances are, it hasn't been your buddy for so long that you got extra candy from yours during that event, so in that case, no, you can't pause it. Under normal circumstances, it's impossible to get the two required candies without walking two uninterrupted 5 KM cycles.

You can, however, finish the 10 KM, then swap it out for the rest of the day, then buddy it again once you're ready to evolve, so you don't have to just keep it going it all day even after walking enough.