Was there a Viking Exchange as well as a Columbian one?

(I just want to add some information, from books by Kåre Prytz, and that does not seem to be mentioned in the other answers.)

According to a note in Latin, written in 1637, based on a chronicle in Skálholt cathedral, which was apparently destroyed by fire in 1630 (Grönlands historiske mindesmærker, vol 3, p. 459), "the Greenlanders voluntarily abandoned the Christian faith and joined the people of America" in 1342.

1342 Groenlandiæ incolæ a vera fide et religione christiana sponte sua defecerunt, et repudiatis omnibus honestis moribus et verus vertutibus ad Americæ populos se converterunt;

This was presumably the few hundred individuals in the Western Settlement, which was found to be abandoned, without any signs of violence or destruction, around that time.

And the settlers of the Plymouth Colony found a grave that does not seem to have been Indian (Mourt's Relation):

When we had marched five or six miles into the woods and could find no signs of any people [...] we found a place like a grave [... containing ...] the bones and skull of a man. The skull had fine yellow hair still on it [...] There was variety of opinions amongst us about the embalmed person. Some thought it was an Indian lord and king. Others said the Indians have all black hair, and never any was seen with brown or yellow hair. Some thought it was a Christian of some special note, which had died amongst them, and they thus buried him to honor him. Others thought they had killed him, and did it in triumph over him.

Prytz thinks there was a mixed population of Norse and Indians in New England in the 14th-16th c. I imagine we might soon find out about that, when enough DNA data becomes available.

The stuff that people usually are talking about when they speak of the Columbian Exchange are domesticated animals, cultivated plants, and diseases.

The first two obviously require people living on both sides to be living at a Neolithic level. In other words, they both have to be farmers or herders.

The third actually requires the same thing, but indirectly. In particular, for a human disease to become endemic in an area, it requires a relatively high population density. The higher the better. The densities required really cannot be achieved without agriculture and/or livestock, and preferably large cities.

This brings us around to the point of contact we are talking about; the Norse and the Native Americans. Here's a map of the cultures in the area. Red is Norse, Green Dorset, yellow Innu, and Orange Beothuk.

This is from the height of the Norse settlement, but the only major thing you'd see from maps with other dates would be the Dorset culture transitioning to Thule (both are ancestors of modern Inuit). The one known Norse attempted settlement point outside what is shown was on the very northernmost tip of Nova Scotia (orange Beothuk territory on this map) about 100+ years prior.

All three native groups belong to different language families, but they shared one thing in common; they were low-density hunter-gatherers. Their lifestyle had no use for Norse crops or livestock, and they had no domesticated crops or livestock of their own to share with the Norse.

The Norse weren't exactly disease-free, but the Greenland colony itself may have been too small for the European disease package to gain a permanent foothold, so those diseases most likely arrived on European trading ships when they did recur. African tropical diseases arriving via this path are obviously right out.

Any diseases the Norse Greenlanders did manage to transmit would likely have burned itself out in the one native camp that contracted it before having a chance to jump to a new one. Which is kind of a shame, because American exposure to European diseases at this early date may have gotten their population crash out of the way early, and given them a chance to evolve some immunity to them prior to Columbus^*. The only known diseases Americans had to contribute were mostly endemic far to the south in Mesoamerica, so the Norse missed out on the opportunity to deliver them to Europe early as well.

^{* - OTOH, that also may have given the Norse the breathing-room they needed to make a go of that Newfoundland settlement, and move into even more agriculturally useful territory further south.}

No. The number of viking transfers back and forth were too small to make a significant difference. We only know from recent finds the Vikings did set up a small temporary settlement in Newfoundland. The discovery was made in the 60's.

It's likely those vikings traveled to Newfoundland to gather wood (almost non existent on Greenland), wintered, and returned. They may have done that a couple of times. Less likely is that they traveled from Iceland all the way to Newfoundland, or even less likely from Norway to Newfoundland.

The settlement on L'Anse aux Meadows was close to the beach. That's where you set up temporary camps. Had they any plans of colonizing, the settlement would would been much bigger and further inland.

Contrast this with Columbus. After his first voyages the area was quickly colonized with lots of ships going back and forth.

Not only did the Spaniards traveled to and from the Americas regularly, they did so in ships much bigger and much better adapted for that purpose.

In their centuries of ocean crossings,

That at least infers there were crossings. I disagree here with you. Most people migrated only once in their lifetime. Norway -> Iceland, Iceland -> Greenland. There was more traffic between Norway and Iceland, but again: not a lot.

I only looked at a possible exchange from the Americas to Europe. After reading @LangLangC 's answer, I have to add that the Viking definitely brought non native animals and plants from Europe to Iceland and Greenland.

The evidence for this is weak, but interesting and indicative of "on a much smaller scale", but not "as well" as in equally transformative:

From East to West:

They introduced the mouse to the American continent. For sure, if we accept Greenland and Iceland as part of that continent, unsure if we only count Newfoundland:

House mice samples from Iceland, whether from archaeological Viking Age material or from modern-day specimens, had an identical mtDNA haplotype to the clade previously linked with Norwegian Vikings. From mtDNA and microsatellite data, the modern-day Icelandic mice also share the low genetic diversity shown by their human hosts on Iceland. Viking Age mice from Greenland had an mtDNA haplotype deriving from the Icelandic haplotype, but the modern-day Greenlandic mice belong to an entirely different mtDNA clade. We found no genetic association between modern Newfoundland mice and the Icelandic/ancient Greenlandic mice (no ancient Newfoundland mice were available). The modern day Icelandic and Newfoundland mice belong to the subspecies M. m. domesticus, the Greenlandic mice to M. m. musculus.

In the North Atlantic region, human settlement history over a thousand years is reflected remarkably by the mtDNA phylogeny of house mice. In Iceland, the mtDNA data show the arrival and continuity of the house mouse population to the present day, while in Greenland the data suggest the arrival, subsequent extinction and recolonization of house mice - in both places mirroring the history of the European human host populations. If house mice arrived in Newfoundland with the Viking settlers at all, then, like the humans, their presence was also fleeting and left no genetic trace. The continuity of mtDNA haplotype in Iceland over 1000 years illustrates that mtDNA can retain the signature of the ancestral house mouse founders. We also show that, in terms of genetic variability, house mouse populations may also track their host human populations.
_{EP Jones et al.: "Fellow travellers: a concordance of colonization patterns between mice and men in the North Atlantic region", BMC Evolutionary Biology, 2012, 12:35, DOI.}

Then there is evidence for livestock, in this case cattle:

The expedition did not stop due to Thorstein’s death. Thorfinn Karlsefni fell in love with Thorstein’s newly widowed wife, Gudrid. The pair rekindled talks of expedition and Thorfinn took Thorstein’s place as the leader. Thorfinn sailed with “sixty men and five women,” and “took livestock of all kinds, for they intended to make a permanent settlement.” This is a significantly larger expedition than any prior expedition. In the first summer, Indigenous people came to the colony. Thorfinn, frightened, locked the gates, but the Indigenous group was able to convince him to trade. When they “saw the milk they wanted to buy nothing else.” The two groups traded milk for furs.
_{Magnus Magnusson & Hermann Pálsson (eds.): "The Vinland Sagas: Grænlendinga Saga and Eirik's Saga", New York University Press, 1966, p57.}

Brigitta Wallace notes that the storage spaces at L’Anse aux Meadows was unusually large for a Norse settlement, and that indicates that it is “a place where goods were collected.” But just no remains of specialised buildings for livestock. This doesn't rule out animals living with humans in the same house or specialised buildings elsewhere. Or just that

The observed effect of these recent fluctuations suggests that winters at L'Anse aux Meadows in the eleventh century were likely snowless and that cattle could indeed have grazed out of doors all winter.
_{Birgitta Wallace: "The Norse in Newfoundland: L'Anse aux Meadows and Vinland", Newfoundland and Labrador Studies. 19 (1): 5–43.}

The L’Anse aux Meadows settlement was intended for year-round occupation as shown in the building construction. All the structures were regular buildings with permanent roofs, not the booths with temporary tent roofs found on seasonally used sites.

Although villages or towns never developed in Greenland or Iceland,20 another characteristic of west-Norse settlements was that a singular farm or estate could not function in isolation. By the time self-sufficient farms were established in Iceland and Greenland, a whole network of settlement was required.
_{Birgitta Linderoth Wallace: "L’Anse aux Meadows and Winland: An Abandoned Experiment", in: James H. Barrett (Ed): "Contact, Continuity, and Collapse: The Norse Colonization of the North Atlantic", Brepols: Turnhout, 2003, p219.)}

If 'exchange' includes trade, then this included all kinds of berries (Vinland) and butternuts from further South. Export of this back to Greenland as a food item is not unlikely; if 'exchange' means bringing species to a new habitat: cultivation of them up North on the other hand is unlikely.

The contact between Norse explorers and settlers with indigenous people lasted for centuries in several small colonies. (Donald E. Warden: "The Extent of Indigenous-Norse Contact and Trade Prior to Columbus" Oglethorpe Journal of Undergraduate Research, Volume 6, Issue 1, August 2016. PDF)

From West to East:

Human(s)?

Analyzing a type of DNA passed only from mother to child, scientists found more than 80 living Icelanders with a genetic variation similar to one found mostly in Native Americans.

This signature probably entered Icelandic bloodlines around A.D. 1000, when the first Viking-American Indian child was born, the study authors theorize.
_{Traci Watson: "American Indian Sailed to Europe With Vikings? –– Centuries before Columbus, a Viking-Indian child may have been born in Iceland", National Geographic, November 26, 2010. Referencing Sigríður Sunna Ebenesersdóttir et al.: "A new subclade of mtDNA haplogroup C1 found in icelanders: Evidence of pre‐columbian contact?", American Journal of Physical Anthropology, Volume 144, Issue 1, January 2011, Pages 92-99. DOI.}

Further, it seems that at least one polar bear (note his natural range) made the way from the Canadian Arctic or at least Greenland via Emperor Frederick to the Sultan of Egypt.

Frederick II’s propensity for exotic wild animals is well documented. Al-Kamil apparently once gave Frederick the gift of an elephant, resuscitating the cultural memory, perhaps, of Harun al-Rashid’s gift of the elephant Abu al-Abbas to that first Holy Roman Emperor, Charlemagne. Frederick arguably topped the Sultan by giving Al-Kamil a polar bear “which to the amazement of the Arabs eats nothing but fish”
_{Numerous quotes available, but this one from p254 of Geraldine Heng: "The Invention of Race in the European Middle Ages", Cambridge University Press: Cambridge, New York, 2018.}

Another was a gift from the King of Norway to Henry the III in 1252.