I had one plant that wash having a really hard time coming up. I poke around it slightly and there were leaves so I figure it would eventually come up.
It rained for several days so all I did was look out the door to be sure the plants were okay. The next time I watered the soil sunk in where the potato was and there were no green leaves. There was a hole about a inch big near the wood on the raised bed. This reminded me of the thread on the disappearing tomatoes. So maybe that was a vole. I kind of think one other plant is missing. I have plenty if they don't get more.
I've had voles attack my potatoes before, but they weren't very interested in the green parts (aside from some chewing around the stalks) they mainly chewed holes in the tubers and had tunnels going from plant to plant.
If you are against setting traps, I can only suggest getting an outdoor cat or setting up some snake habitat.
I only met voles when I moved to NC. I recommend cats and ducks. My cats patrol the gardens hoping for a kill. My first year as a duck owner I found out why I never want to get my fingers too close to their beaks. One morning I found a field mouse dead next to their bowl. It was FLAT. I looked a my Giant Pekin drake and smiled... lovely, they kill mice and snakes!
One of the funniest farm yard arguments here on Dancing Goats Farm, was when I shook out a sheet of plastic that had been forgotten for a week on the ground. Several baby snakes flew out, I yelled and swore and Daisy snatched one out of the air and started running with the 6" snake hanging out of the sides of her beak. Two of the girls took off after her trying to snatch the snake from her bobbing and weaving across the yard. They made a hell of a racket trying to get her to share. Mean while I'm yelling come back there are two more here in the grass. Thankfully Hans did not follow his hens. He called the other girls over and stood guard as they found and ate the other snakes. People had warned me if I put in a small prefab pond for my ducks, I'd have snakes. That day I learned why I'd never seen one.
Good luck! lots of cats and raptors here and we have a plague of voles the warm winter let them breed the whole way though. We have lost some trees, horseradish! and plenty of vegetables to them this year, I'm worried about my strawberries. Keeping grass short and reducing cover for them is meant to help, but I'm not convinced that will put off a burrowing animal and it doesn't seem to upset mine at all.
Traps or poison I think it the only solution, my cat (who died last week :( ) used to catch 2-3 a day and made no noticeable impact on the population, I'm trying some plaster of paris "poison" at the moment, but they've not been interested in the oatmeal at all yet