Blood in vomit or stools means a vet visit. I'm with Bill and Summerrain - call your vet asap.
Eating clean fresh snow is generally harmless; many dogs love it. All mine eat snow regularly with gusto, and other than my little guy with IBD do so without any consequences.
A few dogs, however, may have problems - usually this is because the dog has eaten too much too fast (many dogs get excited and silly in the snow) resulting in natural regurgitation; anything cold ingested on an empty stomach can cause spasms. When dogs regurgitate in such situations the matter is usually clear frothy liquid. If this happens occasionally there is generally nothing to worry about.
But blood in vomit or diarrhea is another thing altogether.
Some dogs may vomit after eating snow because of an underlying gastro-intestinal problem, or perhaps because of something else ingested along with the snow. OR, the snow may be a red herring and the cause of the bloody vomit and diarrhea may be related to something else althgether.
First, could your pup have eaten anything dangerous hidden in the snow? As Bill mentioned, road salt will cause vomiting in many dogs if ingested along with snow. Also, could your dog have eaten a foreign object, frozen cat poo, fertilizer, garbage or something similarly putrid, or doG forbid something seriously toxic like antifreeze spilled in the snow? (If antifreeze is even a remote possibility get to the vet asap.)
Not to get too graphic here, but it's a good idea to examine the type of vomit or diarrhea - what comes out can point to the cause. For instance:
http://www.peteducation.com/article....+2090&aid=3574
If there is blood, whether it is bright red or dark (coffee ground-like) is important information for the vet. When in doubt, I bring a sample to show the vet.
Because I have senior dogs with some chronic health problems I tend to take vomiting and diarrhea seriously, and go the vet sooner than I would with an otherwise healthy dog. A general rule is that vomiting or diarrhea that persists beyond 24 hours warrants a vet visit. But the presence of blood means a call to the vet at the first instance.
FYI, dehydration is something to watch for whenever a dog is vomiting or has diarrhea. To do a quick home check, pinch a fold of skin near his shoulder blades. If the skin immediately falls back into place, dehydration is unlikely. If the skin stays 'tented' dehydration is a possibility and you should call your vet asap.
My mutt with IBD has once had a flare-up shortly after a bout of frantic snow eating, probably a too-much-too-fast thing, but I cannot say with any certainty that snow was the cause. His poor system is so messed up that anything outside his strictly controlled regime can start the cycle. But I would not expect to see this kind of vomiting and diarrhea occurring regularly in an otherwise healthy dog.
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In short, there could be many reasons, especially if this is a one-off incident. But if your dog regularly vomits or has diarrhea further exploration may be warranted, if only to give yourself some peace of mind.
Hope the little one is feeling better soon.
ETA: Cross-posted with you, Sylv. So glad to hear that you've seen the vet, and that your little one is doing better.
As for the winter-stress question:
There is a remote possibility of stress related gastro-intestinal problems. One of mine is a stress vomiter - if something worries him he vomits, usually a clear frothy liquid. This is not all that common, but nonetheless some dogs do react to stress this way. Is your dog the nervous or highly strung type? If something about winter worries your pup, stress could play a part in her tummy troubles - but I would think you'd have seen stress vomiting in other situations before this. Nonetheless, if you think this might be the case I'd treat it as I would any stress situation, by slowly desensitizing her and changing the association. Simply getting her used to cold temps by going for short excursions rather than long hikes initially, playing games so that she begins to see snow as fun, should help her get over this.
My guys who came from warmer climates each needed a little adjustment to get used to 'real' winter; most only showed a bit of surprise at the first snowfall, but one took a week or so to decide that this white stuff was fun rather than frightening. On the other hand, my only Swiss born and bred dog is a total winter wimp - so go figure.
Some of my friends use fleece or waterproof jackets for their short-coated breeds, and tell me that with their jackets the dogs are much happier. As mine are double-coated breeds I've never needed to do so, but you may consider a jacket if your pup seems to shiver or is reluctant to go outside when the temps drop.
But the presence of blood in the vomit makes me think that this was not stress related. Blood generally points to other physical problems.
Continued good wishes to the little one.