It depends on when you started them and how established the root systems are. How about a cold frame? I think that would be your best bet.

I agree, I use a cold frame. As gnomey mentioned, if you are starting deciduous plants, start them early in the year to have a strong enough plant to regrow in the spring. Al

Unfortunately, I don't have a cold frame or place to put one at the momment. What will happen if I just keep them inside through winter?

Most houses inside the air is just too dry during the winter and the moisture in the plant tissue migrates into the air and the plant does not wake up in the spring. Al

My cuttings are not dormant, so will they go to "sleep" or "stay awake" if I keep them warm and lit? The house air does get dry in the winter, would misting or keeping them in a sealed "plant shelf" make any difference?

If you have wild grapevine cuttings that are evergreen, that is not deciduous, Thats news to me. I was discussing cuttings from a deciduous plant which will go dormant in the fall. A dormant plant needs a cool place to sleep with very little moisture around the roots, preferably out of the sunshine. Al

No, they are not evergreen. What I am trying to figure out is if I keep it inside the house will it still go dormant. I thought maybe there was a trigger: temperature, light cycle, etc. that was casual to the plant going dormant. In the absence of such triggers, in this case a warm, bright house, will the plant still try to go dormant or just keep growing all winter long?

It will go dormant anywhere. It needs the rest, and does not make a good house plant. Al