It’s not going to be easy. I would start with a fluid simulation for sure, you’re going to need a ton of tweaking but that’s the baseline method
Welcome 
…
8 years ago:
or for “inspiration”… (?):
i mean… in the end it’s “nothing more” than this:
… but with some different viscosity… and maybe “some pieces” like in the video before.
If you “only wanted it done” then you might better ask in volounteer work… or even may pay for it ?? Because… you didn’t wrote anything about your skill level in 3D in general or blender specifically… 
Usually one has to build up some network of similar interested users before one is getting some ready to use solution… or at least show what one has already tried. So you may also simply try some tutorials and then ask about a more specific problem than “in this genral form”.
( This is actually an advice and no harassment… it’s much easier for anyone to help if they exactly know the context.).
The problem with vomit is that you can’t just do a liquid simulation. Sometimes, there are chunks in there too. Like half-digested hot dogs and whatnot.
You could probably use geometry nodes for that.
This may be so… but for a simplified or stylistic vomit effect like the OP showed in the first image this may be enough ?
And now i giggle about stylistic vomit… is this vomiting with style… 
Ooohh and this user didn’t tell how (s)he did it…
Recommending Houdini Vellum for that. 
Mixing fluid, softbody, grains, hair, cloth … all possible.
Anyway, depending on the style OP is after, I’m not sure simulation is required. Some animated geo might be enough.
What is that? Apple jello?
I’ll say one thing: I’m glad I already ate breakfast before popping into this thread.
Be honest… you just read vomit and directly poped up into this thread 
You know I can’t pass up these opportunities.
You can sim the half-digested dogs and other chunks first, then have the fluid collide with them.
Unnecessary, unless you want to have the character explode - wait 
Vomit bomb sounds like an awesome special move.