While it's difficult to prove a negative, and there's no telling where Nisio Isin got his inspiration from, I would be willing to claim that that most of the apparitions appearing in the Monogatari series aren't quite based on real-life stories.
Broadly, one point in favor of this position is that other people on the internet (in Japan) have also tried to find origins for the various apparitions and failed. Given the popularity of Monogatari, this strongly suggests that there indeed aren't any real-life bases for most of the apparitions, though, of course, this doesn't amount to conclusive proof.
Hitagi's crab is a difficult case to handle. Crabs are a semi-frequent feature of Japanese folklore - see, for example, the tale of the Crab and the Monkey (which Meme mentions), or the legends surrounding the Heike crabs, so we can't discard out-of-hand the possibility that the "Weight Crab" (omoshikani) has basis in actual legend. Nonetheless, it seems improbable.
Meme claims that legends of the "Weight Crab" originated from peoples living in the mountains of Kyushu. However, he goes on to point out that the areas from which these legends originated - the mountainous parts of Oita and Miyazaki - are not actually places where you will find any crabs. Meme brushes this off, explaining that it's easier to concoct grand legends about crabs when you don't have any actual crabs at hand.
I'm not convinced that this really makes sense in a real-world context, though - do people tend to feature in their legends creatures that they have little contact with? I'm no anthropologist, but it seems unlikely. Someone familiar with the folklore of Kyushu would know more, I'm sure.
There are also some things about Hitagi's crab that just seem too convenient. For example, its name is incredibly punnable in ways relevant to the plot - 思い omoi "thoughts" vs. 想い omoi "feelings" vs. 重い omoi "heavy". The resolution of this arc is strongly related to the realization that the crab took not both Hitagi's weight, but also her feelings from her. It would be a damn surprise if there was an existing legend out there that matched up with this. Far more likely is that Nisio Isin thought that this would be an amusing pun and invented a legend to go along with it.
The points I've made in this section also generally apply to Mayoi's Snail and Nadeko's Constrictor.
Again, it's difficult to make a strong claim that Mayoi's snail has no basis in actual legend. Snails are not prominent features in Japanese legend, though.
The "Monkey's Paw" does have a basis in real-life - a short story by W. W. Jacobs titled as such. And that's the problem - Kanbaru's apparition wasn't a Monkey's Paw, but rather a Rainy Devil, which is an apparition possessing only superficial similarities to the Monkey's Paw of Jacobs's story.
It's pretty clear here that Nisio Isin was deliberately playing with the reader's expectations by having the characters themselves incorrectly identify the arm as a Monkey's Paw - and then they go to visit Meme, who pulls the rug out from under them, revealing that it was something else altogether. This strongly suggests that the Rainy Devil is a creation of Nisio Isin's.
To a certain extent, it's very difficult to show that the Constrictor (jagirinawa) has no basis in legend, since there's nothing terribly unique about it - it's a snake that possesses people. That's basically all there is to it. Snakes are certainly common features of legend throughout the world (as Meme points out), and I wouldn't be surprised if some group of people somewhere in the world did believe in a constrictor snake that possessed people.
Like with the crab, though, this is most likely a case of the author taking a motif well-established in folklore and putting an original spin on it.
I'll update this answer later with more details. A few notes in the meanwhile:
- Hanekawa's cat is certainly an invention of the author, as you point out.
- Shinobu explicitly identifies Tsukihi's phoenix (the shidenodori) as having some unusual features that don't match up to the conventional conception of a phoenix.
- Given that Karen's bee is a fabrication even in the context of the story, I'd imagine legends of it don't exist out here in the real world either.
And for Monogatari Second Season:
--There are no tigers in Japan. And besides, Hanekawa's tiger (the kako) is far too punnable to be real, just like Hitagi's crab.
--Jiangshi are a real thing, though Monogatari jiangshis appear to differ from real-life jiangshis in some minor ways.
--Nadeko's kuchinawa might be inspired by snake legends, but it's pretty obviously original for the most part.
--The apparition-eating darkness of Onimonogatari only makes sense in the context of the Monogatari world. It wouldn't make sense for it to be based on a real legend.