Vanishing points are a simplified concept that artists use to help them draw perspective images of idealized pinhole cameras. Like so much in art history it's only one possible way to draw 3D imagery. You can use other methods but they are more convoluted and not widely taught to people. Artists have a person in the loop so that person can use judgement so it's the quick and dirty way is good enough.
There is no need to know or even draw a line to a vanishing point to draw a correct perspective image. You can also use advanced methods to draw fisheye perspectives where straight lines are no longer straight.
- A vanishing point does not exist in 3D space ($R^3$).
- A vanishing point exists in a projection $R^2$.
- But it does exist in a homogenious projective space. But needs to be modelled separately from normal geometry as you can not rely on taking it with you form 3d space. Like normal points
Therefore you can not transform a vanishing point from 3D to 2D which is what you originally asked. But you can create a model where its true.
Mathematically such a point exists at the limit of an infinite line, but there's no set where they overlap. Intersects at infinity is OK. But in practice, there's no difference between a mathematical point and an apparent point. They transform the same way in spaces they exist in.
Now, it is important that we do not regard a 2D image consisting of pixels. We might be rendering vector graphics and thus computing points coordinates not pixels. Yet for practical reasons, an imaging system has to have some minimal width for a line for it to be visible.
Again, as I said earlier, any point sufficiently far away from the camera in 3D is at the mathematically approximate location of a vanishing point in 2D. For all intents and purposes, this is a vanishing point. This point behaves like any other point. But is this in any way useful for your line of query? Probably not since this definition would mean that there are lots of vanishing points in any image, horizon line being all the vanishing points for all lines of a plane.* Your line questioning is sufficiently close to a XY problem to never have a good answer. Consider asking what you want to achieve instead.
Is a vanishing point real? This is a really deep metaphysical question**. In some sense, it's not, if you give me the ability to zoom infinitely then you will never reach it. But you can still identify it and see it. It both does and does not exist at the same time, it's a concept. There is no contradiction here just different models that have different results. The concept exists, after all, it has a Wikipedia page, but so does spider man. Real-world is like that, with lots and lots of conflicting models.***
* If you haven't noticed it, is not very fruitful to ask: "Where is the vanishing point of this image, this camera, or this model?". Because there are probably too many of them. It is however much more fruitful to ask: "Where is the vanishing point of this line?"
** ask a philosophy.se?
*** this reminds me of the fractal nature of the world. If you ask how long is the coastline of a country is you get different answers depending on how accurately you measure. The length of this may be infinite. Which admittedly is not very useful and probably not accepted as real in any pub quiz.