We are visiting Cinque Terre in January. We would like to stay in Vernazza but will arrive in the evening. Are any restaurants open in the winter?
Very few. It is very much the off-season. It is a wet time of year. There are far better choices in February.
Echoing Laurel, I wouldn't choose to visit the CT in January as the majority of businesses will be closed. There will be a restaurant or two still open but I'd choose a different region/location if wishing to have a bigger choice of places to eat, and a choice of indoor activities if the weather is miserable...which it could be in the dead of winter. I guess it depends on what you want to get out of your visit there, and how willing you are to venture out if it's windy, wet and cold?
I would skip it altogether. At most visit it on a day trip, if you happen to stay in Florence, or even better from Lucca, if part of your itinerary.