This is a quick reply off the top of my head, without checking references.
The Eldar are certain to know of the Valar, and I should guess they know of all of them, even if naturally not all Eldar met all Valar. In the case of Avari, they also knew of the Valar, and even if they never came to Valinor while alive (if slain or in other way disembodied they would go to Mandos regardless of what type of Elf they might be) I suppose the knowledge was passed on down the generations in one way or another.
Dwarves have their own stories about how they were created, but it is strongly suggested that they knew Aule to be a Vala (or at least a Power of some kind), therefore I expect them to know of the Valar, even if they know of only one at least.
Men did not know about Valar before coming to Beleriand in the First Age of the Sun, but there they learnt this from the Elves. In the Silmarillion it is said that Men also go to Mandos when they die, and thence leave the circles of the world forever (and only Mandos and Manwe know where they go). Since Tolkien envisioned his world to be set in our real timeline (in the distant past) you can only speculate to what degree the men living today all over the world know of the Valar (or should we rather say, more conventionally, angels), and to what extent they believe in them. Hobbits are essentially human, so I include them here as well. Although by the time of LotR, I have no idea if the average hobbit knew about the True West and the Powers there.
In all of the above, bear in mind also that many Elves, Dwarves and Men saw or heard about Melkor, and he too was a Vala.
There certainly are passages throughout Tolkien's opus which may better emphasize the issue you brought up; I might look them up if no one else does in replying to your question.
I would say that the men of Numenorean descent knew about the Valar, which are steeped in the history of that people. The people of Rohan knew of them as well I believe, though must have had their own names for them.
As for hobbits - I can't imagine the majority of them knew about the Valar. They struggled with concepts like orcs and dragons. They knew about elves leaving 'for other shores' but not sure how much further it goes than that.
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Utúlie'n aurë! Aiya Eldalië ar Atanatári, utúlie'n aurë! Auta i lómë! Aurë entuluva!
I agree with what has been posted already about knowledge of the valar among dwarves and elves.
Certainly there seems to be one "school of thought" among elves and the men their lives in middle earth influenced and perhaps another among dwarves (although these aren't necessarily contradictory - differences may be a matter of emphasis, relationships to various deities, and differing locations of holes in the full theory of it).
I'd assume that both dwarven and elven "lines of tradition" would probably include Yavanna in some way.
As far as what may be known by other communities (especially the human communities in the south and west) this is a matter of speculation. It is likely that something was explained about it by Sauron but it was probably far from true. Similarly, there may be other more isolated human populations here or there who could have had a visit from a stray elf, dwarf, maia or other such being who may have taught them this or that which may or may not include Yavanna (depending on who the population contacted, how they passed down information, and the nature of the information they gathered from the encounter). But such possibilities are just in the realm of imagination since there isn't much (if anything) said about it explicitly as far as I can remember.
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There is nothing like looking, if you want to find something. You certainly usually find something, if you look, but it is not always quite the something you were after. -J.R.R. Tolkien (The Hobbit)
