I approached this topic in
this chapter of my Amerindian-themed TL. To summarise, the issue is the Norse are so far from Europe (the source of plagues) that it would be incredibly hard to transmit illnesses except for those with a very long incubation period, high contagiousness, and asymptomatic transmission. That's pretty much just chickenpox (via shingles), mumps, and whooping cough. IIRC typhus and other flea-born illnesses
could spread similarly but introduced rats will spread very slowly, especially since the Norse will bring cats along with them too (although to be fair, cats prefer to munch on native rodents instead of introduced rats/mice). I disagree with the idea other diseases could spread given the history of epidemics in medieval/early modern Iceland. If they did, they'd be very local and burn out quickly. Even my own TL is pushing things with diseases, which relies on larger populations, mobility, and connected economies brought about by from domesticated caribou and mountain goats.
These diseases will be lethal, in particular chickenpox since it's fairly dangerous to adults who contract it. Whooping cough increases infant mortality and mumps mostly causes infertility in men (and to a lesser degree women) along with occasionally killing. This has an obvious effect on demographics. Amerindian knowledge of disease is vague--archaeology suggests that some towns appear to have been abandoned because they were too large and the population had insufficient knowledge of sanitation. While they suffered from colds and spread forms of mild
Treponema diseases (akin to yaws and pinta) and in some regions tuberculosis, there were no real diseases both easily spread and very dangerous. So these diseases arriving would be quite a shock, and they very well could spread across the Americas given they spread from Iceland or Greenland to the New World.
The effect is the natives may culturally re-evaluate diseases and epidemics. Chickenpox will likely be endemic in most of the Americas because of shingles, although the other two diseases only in isolated parts (Mesoamerica to the Andes, maybe the Mississippian heartland) thanks to lack of population density. This is still good in the sense it gives some cultural mechanisms to respond to an epidemic and gives the population resistance to at least one disease that OTL they had none. OTL many people in the Americas died of starvation because multiple diseases struck at once and disrupted seasonal hunting/gathering/fishing. We could expect to see the same TTL, but without European interference. Hunter gatherers will suffer the most, leading to huge population shifts.
Culturally this will lead to major changes. It would not be pleasant to be in a major city of 12th/early 13th century North America TTL, given it's likely half-starved semi-nomads in nearby areas will be up in arms, spurred on by prophets and religious revival. These people will bring warfare and destruction out of the sheer need for survival. To compound, in Western North America and much of Mesoamerica there was a major drought in the middle of the 12th century. Historians would reflect that epidemic disease helped bring down Cahokia, Chaco Canyon, Tula/the Toltecs, El Tajin, Chichen Itza, etc.
They probably did just not small pox, it's too bad they didn't stick around long enough to deliver a breed able population of horses, some ironworking, and domestic animals.
That was enough time to change things enough that the natives could have stood a better chance.
If they brought disease, then it never spread beyond a few indigenous villages in Vinland/Markland/Greenland.