No, this is a hoax. There are several very large red flags in their story. First, none of the artifacts are from any sort of archaeological context. The only way we know where any of them came from is a long, convoluted, "my friend's cousin knew a guy who saw some people pull it out and then they kept it hidden for years" story. There is no way to prove the sword came from where they said it did. Second, they tell you just enough sciency info to make you think, but XRF doesn't quite work the way they say it does. All it tells you is the composition of the metal that the sword is made from. They supposedly tested it against another sword, but they don't tell you where that sword came from, so for all we know, both swords are modern. Third, you can buy exact copies of this sword on Ebay for like 75 euros. Or from this site a few years ago. Plus, the patina on the sword is all wrong for being underwater for over a millennium.

And that's just what's wrong with the sword.

For the other artifacts, they don't have any provenance either (we have no idea where they came from), and it's very common to purchase antiquities abroad and bring them to the States. Sometimes they get lost. I literally know people who have done this and lost Roman era coins in the United States. Unless the item is from a legitimate archaeological excavation and has been properly recorded, you have no idea where it's from.

Long story short, this story is almost certainly a hoax. Only if (and it's a very huge if) this mysterious white paper is published and it answers these questions of provenance, as well as having the names of established scholars in the field, and survives a rigorous peer review process, then will the theory have any weight whatsoever. The fact that this is coming out as a white paper and not in a published peer-reviewed journal is not a good sign, and shows that they don't have confidence that their work will stand up to scholarly scrutiny.

I should preface this by saying that the information in the article is insufficient to make a definitive claim, but based on what is available I would be highly skeptical of their claims. Extraordinary claims require extraordinary proof, and in archaeological terms it looks like they are offering very marginal evidence for their extraordinary claims.

The most important component of evaluating archaeological evidence is the context of the find, and sadly, without examining the report itself, we can't really evaluate that from this article alone.

The three things I would want to see to really seal the deal here is solid stratigraphic context for these artifacts, absolute dating, and solid association between the artifacts.

For the stratigraphic context, I mean the position of the objects in the ground relative to other objects and geological features. This is the basis of relative dating (in contrast to absolute dating which I will discuss below) which is the proposition that certain artifacts are older than artifacts above them in the ground and newer than artifacts below them in the ground. In this case, there should be a fairly well established sequence of indigenous artifacts that we could use to date these Roman artifacts. For instance, one style of stone tool found below the objects and another (later) style of tool found above them. This would allow us to give a relative range of dates for the objects and evaluate the proposition (based on those dates) that they are Roman.

That said, it sounds like from the article that these objects were all found on the surface which is a huge red flag. Certainly you can find objects that old on the surface in many places and these can be used to support an argument, but we would need to evaluate the geological processes in that location to determine if objects this age are likely to be found on the surface. Again, not enough information in the article for this, but forest ecosystems on the East coast like this one tend to have a high rate of soil formation and deposition. Being an island might affect that, but my default assumption would be that so many finds on or near the surface in that kind of context suggests they are much more recent than Roman explorers. Again, can't say definitively with the information presented.

Secondly, absolute dating. If they want to claim great antiquity for these objects they really need to prove it either with relative dating (as I described above by having solid stratigraphic context in the ground) or with absolute dating techniques (that provide a calendar year or range of years) like radiocarbon. Of course, most of what they found is metal and so unusable for radiocarbon dating, but they can use objects in strong association with these metal artifacts. For instance, if they had coins in a hearth (and they felt confident their hadn't been any disturbance) they could date the charcoal in the hearth and by extension the coins found within. Again, it doesn't appear that they have any strong context for any of these artifacts that would allow this kind of absolute dating or even the relative dating I mentioned previously.

Finally, association among the objects. While they are proposing a fairly ephemeral event that wouldn't leave much in the way of material evidence, we should still expect to find more than just one object out of context with any others which is what it sounds like they have. For instance, instead of finding a sword on one side of the island, some coins elsewhere, and a shield on another side, we should ideally find some of these objects in the same local. What that would mean is that we can have greater certainty that they were deposited by the same event (e.g. Roman explorers). What we have in this case is a number of artifacts that may or may not be related to each other at all aside from vague stylistic concordance. That said, I am assuming this from the article which doesn't go into great detail about the locations of the finds.

So, in summary, the kind of strong evidence we should like to see in order to make their argument credible is entirely lacking, as much as I can tell from the article. Instead of having several objects in association with each other, in a subsurface context that can be dated either relatively or absolutely, we have a collection of disassociated objects without much potential for dating them. That is about the weakest kind of archaeological evidence you could ever present, yet they are utilizing that weak evidence to support a very extraordinary claim. This doesn't disprove their claim (but then again, it is almost impossible to disprove anything archaeologically), but it isn't even close to the airtight argument the researchers make it out to be in the article.

Edit: I should also add that while they claim these are all Roman objects, they don't bother to date them in any way using their stylistic attributes. This is fundamental archaeology, that material culture changes over time and we can track that change. For instance (and I don't know if this is true, but as an example) the style of gladius they found might date to a certain period while the style of coins and shield date to yet other periods based on known changes in how these objects were made within the Roman period. The coins especially should be easily dateable, yet they don't provide any of this evidence. If they had a strong case, all these objects should date very closely to all the others based on this stylistic change.