I live in a small mountain village at 1800m altitude in the Swiss Alps, so there is snow most time of the year. It can snow in October of November and the snow sometimes stays till the end of May/June.
So do I still run? Yes! Also in winter you can still run, but the options are limited. In a lot of villages they prepare hiking trails, which means you can run on them without the big snow-shoes. And the cross-country slopes are well for running. Because of the snow you have to stay lower, but there are still plenty of possibilities left to go out for a run!
If there is a lot of fresh snow, I like to take my snow-shoes and try to snow-shoe-hike up as quick as possible. Which is hard I can tell you! But besides an aerobic training you will train strength as well, because it can be pretty hard to make a new track in a big layer of fresh snow.
Two other favorite sports of mine are skinning up with my skis or go cross-country skiing. The nice thing about those sports is that you are doing a whole body training. Besides training your legs you also have to use your arms and abdominals a lot. Especially by the cross-country skiing they are very important for moving forward.
I also think that changing to other sports activities during winter gives you a very good base for trail running during summer. You can keep your physical condition high and because of using other muscles you will train your whole body during winter. And that will give you lots of advantages during your big runs in summer!
Siebrig