I understand that some historians argue that 1700 rather than the more conventional cutoff of circa 1500 should be considered the end of the Middle Ages. Who are they and what is their argument?

Broadly speaking, if this is a thing (which I'm sure it is somewhere), it's a niche view. Most of us Medievalists chug along comfortably ending our period somewhere in the 1400-1500 area, with a smaller number stretching as late as 1550 or so, and some going as early as the 14th century (See u/rebelolemiss above, for example, though from their description of their work, Renaissance or Early Modern Scholar might be a better description, though I obviously don't really know).

I can't really argue the point myself, because I don't have in front of me and cannot find anyone specifically arguing for 1700 as an end to the middle ages, but I think I can guess at where that's coming from, and talk a little about the historiographical trends that might lead someone to say this.

So there's an ongoing debate about how much the early modern period represents a disjunction from the middle ages and how much it represents continuity. As a rule, Medievalists generally tend to be more comfortable saying that the trends in the middle ages continued on into the modern period, and that the modern period represents a smooth continuation of late medieval trends, while Early Modernists and Renaissance Scholars tend to be more comfortable drawing a sharper distinction. Generally, what this means, is that the latter tend to push the date a little earlier, to emphasize the radical change happening in the 14th and 15th centuries, while medievalists tend to push the date later, to emphasize the ongoing practices and trends.

To me, 1700 sounds like a bit of medievalist cheek, with a grain of truth in it, similar to a point made in a book I do have in front of me, claiming that Italy only became truly Medieval in 1527, with the Sack of Rome, Hapsburg rule, and the Counter Reformation. The point is, there are ongoing trends, increasing centralization and density of government, migration to the cities, increasing agricultural surplus supporting more sophisticated and specialized production, increasing education and literacy, etc that start in perhaps 1000 and more or less keep going. To what degree this consistency is something you see as more important, versus the radical shifts of certain seismic events like, say, the Reformation is really going to effect where you draw the line. Hell, if you wanted to, you could push the middle ages almost to the 19th century, with the fall of the ancien regime and such. Fundamentally, what we're doing when we draw these arbitrary lines is trying to make this sort of point, and as I see it, claiming 1700 as the end of the middle ages is just an extreme way to do it.

For my part, I'm a medievalist, and definitely see a pretty strong continuity between the Middle Ages and the Early Modern period. I'd certainly disagree with u/rebelolemiss's characterization of Petrarch as the beginning of the end of the Middle ages (though I suppose Petrach would probably disagree with me on that), and I'd honestly be comfortable pushing the middle ages pretty close to the 16th century, maybe even a little beyond in some places (Scotland comes to mind), because from my perspective, printing's a big shift, but it takes a little while to really get off the ground, and I fundamentally see it as an acceleration of ongoing trends, not something totally new, so I don't feel bad about overlapping with the rise of printing a little bit, though it's a big enough change that I can't ignore it.

The only other option I can think of is they were talking about outside of a European context, which is fair, but I am unqualified to speak to that, so I'll leave it out.