We will be staying in town for a couple of days. Are there any wineries within walking distance?
I once tracked down and visited a very unique winery in Montefalco--Antica Azienda Agricola Paolo Bea. Its on a road leading into town and is in a new facility making their signature sagrantino as well as other wines by extremely traditional methods--whole cluster, organic farming, gravity feed, natural fermentation. I'm certain you could walk to it from the center of town and be prepared for a very passionate presentation.
Between Montefalco and Bevagna, down in the valley, is Arnaldo Caprai a very sophisticated vintner with classy wines and facilities but you would have to taxi or drive even though it is very close.
I think that walking some kilometres on the road Foligno-Montefalco is basically possible if one likes to walk (let's say, to hike). There are not many cars and the road is not dangerous. It also depends on the weather and on the season; I would do it in a fine sunny day, I wouldn't do it in the middle of the winter with icy cold and little daylight.
Finding a taxi in Montefalco is not easy at all, too.
As Aristofane says, it depends upon what you mean by walking distance. In the late 90's we visited Azienda Agricola Adanti. We had a very enjoyable afternoon learning about and trying their superb wines.
Have look at google maps.