Number one problem--why should they? What does Vinland offer which the Vikings can't get elsewhere? This is a common topic here, but any real PoD involves much more complex American Indian societies for them to trade with. Yes, it's about the height of the Mississippian period, but they'd need to go as far south as the Carolinas to actually find natives worth trading with.
But successful Vinland isn't hard. The Beothuk natives were no more than a thousand people. Norse on the Avalon Peninsula (the best part of Newfoundland) could drive out the locals entirely from the area and secure the peninsula with some forts. They'd find the Grand Banks to help feed them in addition to other means. Whatever disease they'd bring over (distance made transferring epidemics to even Iceland challenging) would do the rest, assuming the Norse didn't destroy the Beothuk first.
Their main influence would be their trade with natives, where they'd give them farm animals, European crops, and all sorts of iron tools in exchange for furs and whatever else. Newfoundland in the Medieval Warm Period can sustain quite a bit of people comfortably, and settlers from Iceland and Greenland won't be picky. The best trade goods would come from far south--yaupon tea was a major American Indian trade good, traded from its native range along the coast to as far north as Cahokia. This caffeine-containing tea, favoured by American Indians for religious rituals, would find favour with the Norse. Norse products will be a major trade good for all coastal American Indians, so it's likely Norse expansion takes place as part of small trading posts along the coast.
Let's say they are mostly immune to the Old World diseases by this time.
This will not occur, since even OTL, Old World diseases decimated the Plains Indians and PNW Indians all throughout the mid and late 19th century. What you'd ideally get is disease wrecking the Mississippians in, say, the 12th century when the Norse make their trading posts there, and instead of the increasing decline they had in the 13th century onward, they rebuild in a stronger fashion at that point. This means a lot of OTL Indians are much stronger, and anyone visiting OTL Kentucky or Tennessee would find a thriving civilisation (unlike OTL where the locals abandoned the place and it was used as a hunting ground by various outsiders). The animals brought by the Norse combined with their tools would likely prevent the decline of the Mississippians, and perhaps stimulate them to new heights, especially when they start mining the gold in the Carolinas and Georgia.
At this point, Vinland is likely to be relatively independent, and perhaps can raise up its own king in opposition to any Norwegian or Danish king who would want to control its revenues.