Is there a notion of causality in physical laws?

Firstly, "an equation that describes a definitive outcome as the result of some input factors" is a description of a deterministic relationship between input and output, not needfully a causal one. Causality and determinism are not the same. What if the equation is invertible, so that one can swap the roles of input and output and describe the new equation in the same way?

Secondly, although it is hard to second guess what a writer is getting at, I don't think the denial of all notions of causality is quite as strong or deep as one might think from a first reading of this passage. I believe the writer is getting at something like the following. There is a large body of wholly experimental knowledge observing that the phases or sub-events within certain physical processes always have the same time ordering, that, for example, one always has to cook eggs before one can eat cooked and not raw eggs. I have never observed this process play out in anything other than this order. This observation of a fixed time ordering of this particular process and many others is all that we can rigorously say in describing a notion of causality. I discuss these bare observations and their role in a special relativistic notion of causality in this answer here.

What the author is most likely denying is a rigorous notion of a unifying, unambiguous notion of a causal relationship beyond these experimental observations. What defines notions like causal agent, effect and other supposed abstractions? When the author denies "causality", I think he/she is saying that a rigorous definition of these more abstract notions is likely impossible, and is probably the result of our projecting our humanly (but probably impossible to define rigorously) notions of will, wish, purpose and other teleological ideas onto physical processes. No philosopher, mathematician or physicist has so far been able to do this in an accepted, universally accepted way.

So, in short, I don't believe the writer is denying the existence of causal filters or other fixed-time-ordered processes. He/she is simply saying that a rigorous definition of our human-centric ideas of causality is probably not possible and warning us not to expect that any rigorous notion of causality should resemble our human-centric expectations.

What we have in physics is experimental data and models attempting to describe the correlations found in the data. Then there is the more or less subjective and cultural meaning given to the models, often implicitly so.

For example, suppose a law of physics showing that Earth orbits around the sun. Let's say such a law, based on the observation of the sky which shows consistent patterns (correlations), based also on earlier models having identified the sun with a specific object located in space (as opposed to, for example, a round patch of light glued to a celestial sphere), describes sun and Earth as being attracted in proportion to the product of their masses and in inverse proportion to the square of their distance. By manipulating this relation of attraction, one can show that Earth perfoms an elliptical orbit around the sun.

Where is the causality here?

Is it that the sun, because it is much more massive than Earth, is the "attractive cause" of Earth motion? That's how some people would make sense of the law, but note that this interpretation is no physics per se, but arises from the way people relate to the physical law by giving it a meaning. And in that case, a physicist could easily correct them by saying that Earth also attracts the sun.

Is causality part of the whole Hamiltonian picture: setting up the initial conditions of sun and Earth at specific points and with specific velocities, the law itself (translated in its potential form) is the cause of the evolving orbital system? Not quite, because, as other answers and comments pointed out, determinism is not causality: if the equations are time-reversible, the final state of the system is as much the cause of the initial state than the other way around, at least physics-wise. For an ordinary person such as myself, since time seems to flow from now to the future, it is the initial state that is the cause for future ones. But again, this is just me, not physics, speaking, trying to make sense of the equations.

Causality implies a narration: this made that happen.

In physics, a narration always comes from an exterior conceptual framework: "I dropped the egg which broke" is a narration consistent with physical laws, but if we wanted to study what it is this sentence talks about, we would have to undertake the description of the egg, the ground, the gravitation field, using statistical quantities and finally invoking the second law of thermodynamics to explain that "because the egg hit the ground it was irreversibly smashed", and during this process some operations of simplification and coarse-graining have to be made, by which narration is injected into the (ideal) bare and raw physical description of the system, which is timeless and eventless, only relational.

There has to be. The only way we test laws is by doing experiments. If causality was not there then we would not have been able to prove any law experimentally.

Doing actual calculations is a different story - Some processes are so random and complex that not only they appear to be non-deterministic, they practically are that way ("for us"). For looking at such process, we do not need to go quantum. We can not even calculate what day/time a particular molecule of water will evaporate from a swimming pool. Even though we understand the process of evaporation quite well.

"For us" means that the the complex and random processes are non deterministic for us, the observers, but not for the nature. The nature would be fully deterministic. Actually the randomness is caused by determinism of nature itself.

If causality was missing, we would not be able to experimentally prove any physical law. Simply because the cause and effect - "You do this experiment, and you will get this result" would not work for those laws. Any law itself may be defining causality in some way.