Well when my son was that age, his teams basically played all year round. Though it did slow down a bit around the holidays, it picked by up just after the first of the year. There is a facility just a few miles from us that hosts youth baseball tournaments. There is a tournament there just about every single weekend of the year, And they are not alone. You can always find someplace to play your team. During High School baseball season --which starts first week of February -- is the only time things get shut down at all, but only for HS age teams. The pre high school plays all through the year. And many, many high school players will play with their tournament teams right up until the season officially starts.
Here's the schedule of tournaments at the nearby facilities for one of the main tournament organizers in this area from after Thanksgiving though January (and this is just for one of several tournament organizers in the area)
Now different teams have different philosophies. Those teams that are serious about preparing guys for the next level or protecting young arms do tend to shut it down for at least a couple of months. But there are SO many travel teams in this area especially at the pre-high school level, where coaches are basically out to be make a few bucks -- well, more than a few bucks -- that if you want to play all year with somebody, you can definitely do so.
But I gotta say, this area definitely isn't So. Cal or Florida, where I imagine weather is almost never an issue. On a January morning, at 8am, it can be damned cold out there, even if there is no snow on the ground. And I used to wonder, why so many teams try to play all year round and whether it was really about developing the kids or the almighty dollar.