In English grammar, I know about the concept of vocative case quite well — thanks, John; I know, Sarah, you are going tomorrow etc. But my query: should we use a comma or not with everyone? For example, Happy New Year, everyone. I personally think a comma is required — but then, still want to make it clear.

Would the presence or absence of the comma make the sentence any clearer?

Well, for me, more than clarity, it's about following grammar rules. Hence, I want to clear this doubt.

A comma is used to keep separate ideas in a sentence separate. A good example of this is:

I know Sarah…: Sarah and I are acquainted

I know, Sarah,…: I know something

ALWAYS use a comma with such ( except with appositives:)

...appointed to Acme Industries. John Smith, CEO stated…
CEO John Smith stated...

Claire, I am acquainted with those rules — I just want to clarify my everyone (vocative case) doubt. I think you hinted at using a comma in that situation — if I am not mistaken.