This doesn't sound to me like a behavior problem, it sounds like a serious health concern. Talk to your vet about it as soon as possible and see what they have to say. My instinct is that there is some toxin at play, but I am not trained in vet medicine at all and may be wrong. Still, it's at least worth testing for and thinking about before ruling it out.
Is there currently salt or other ice-melting chemicals on the roads or outside your home? If so, is there any way at all that your dog could be ingesting some of them? Does he go for walks on salted roads? Do you walk on a salted sidewalk and then bring your boots inside? This issue is quite common and would likely cause the problems that you described. The solution is mostly based in awareness and vigilance- you can wash his paws in warm water after a walk, remove your boots before coming inside or clean the floor thoroughly if there are any traces of salt.
If it can't be outside chemicals, is it possible that your pup is eating, chewing on, or licking some winter-time item that you have in your home? My family used to decorate with fake styrofoam berries that attached to various surfaces with twist ties. They were subtle and beautiful and we put them everywhere and when they started disappearing we assumed the cat had stolen them and lost them under rugs/the fridge. Until we found my dog chewing on them. She liked the texture of the foam, but the paint was toxic and made her quite sick. Is there any decoration that your dog could be licking when you're not looking? Do you have pine scented things in your house that your pup could be allergic to? Is it possible there is chocolate or other toxic foods lying around in easy reach? Do you have houseplants that he could be chewing on or licking? (Poinsettia is toxic to dogs, for example) Do you have clothing items or something else that might be coated in toxins (e.g., fire starting logs, waterproofing on boots)?
The next time your dog has diarrhea or vomits, save some of it in a plastic baggie or air-tight container and take it to the vet. They can do an analysis of it and find out if there is anything suspicious that could be causing the problem.
My second guess would be allergies, although this is not the most common manifestation of them in dogs and most things that dogs are allergic to are not present only in wintertime. But as I mentioned above, it is possible that there is a reaction to scents or decorations that you have out this time of year (candles, plants, woodsmoke, etc.). Think through things that are in your home now that are not typically present. Your vet can give you advice on if allergy testing might be something you would want to look into.
One final thought would be food. Are there any dietary changes that happen in winter for your dog? If you feed a consistent diet of one brand of dog food, then this is likely not the issue at play. But if you feed raw or homemade food, then it is worth thinking through where the meat is stored, where it comes from, etc. Is it possible that it is going bad? Alternatively, do you feed your dog table scraps? Holiday food tends to be rich and complex, moreso than food the rest of the year. If you allow your dog to chew on animal bones, there could be small sharp pieces breaking off and being ingested (although you would likely see blood in the feces and/or vomit).
I wish you the best of luck in figuring this out!