Disclaimer: I am not a chemist by any means, and I only have knowledge limited to what I learned in my university's Chemistry III course. Basic understanding of everything up to valence electron orbitals.
Why is there no set of rules to follow which can predict the product of chemical reactions? To me, it seems that every other STEM field has models to predict results (physics, thermodynamics, fluid mechanics, probability, etc) but chemistry is the outlier.
Refer to this previous question: How can I predict if a reaction will occur between any two (or more) substances? The answers given state that empirical tests are the best way we've gotten to predict reactions, because we can discern patterns or "families" of reactions to predict outcomes. Are we only limited to guessing at "family" reactions?
In other words, why am I limited to knowing my reactants and products, then figuring out the process? Can I know the reactants, hypothesize the process, and predict the product?
If the answer is "It's complicated", I would enjoy a push in the right direciton - like if valence orbitals actually do help us predict, or any laws of energy conservations etc, please give me something which I can go research.

First of all, I'd ask: what do you admit as "chemistry"? You mentioned thermodynamics as being a field where you have "models to predict results". But thermodynamics is extremely important in chemistry; it wouldn't be right if we classified it as being solely physics. There is a large amount of chemistry that can be predicted very well from first principles, especially using quantum mechanics. As of the time of writing, I work in spectroscopy, which is a field that is pretty well described by QM. Although there is a certain degree of overlap with physics, we again can't dismiss these as not being chemistry.
But, I guess, you are probably asking about chemical reactivity.
There are several different answers to this depending on what angle you want to approach it from. All of these rely on the fact that the fundamental theory that underlies the behaviour of atoms and molecules is quantum mechanics, i.e. the Schrödinger equation.*
Addendum: please also look at the other answers, as each of them bring up different excellent points and perspectives.
(1) It's too difficult to do QM predictions on a large scale
Now, the Schrödinger equation cannot be solved on real-life scales.† Recall that Avogadro's number, which relates molecular scales to real-life scales, is ~$10^{23}$. If you have a beaker full of molecules, it's quite literally impossible to quantum mechanically simulate all of them, as well as all the possible things that they could do. "Large"-ish systems (still nowhere near real-life scales, mind you — let's say ~$10^3$ to $10^5$) can be simulated using approximate laws, such as classical mechanics. But then you lose out on the quantum mechanical behaviour.
So, fundamentally, it is not possible to predict chemistry from first principles simply because of the scale that would be needed.
(2) Small-scale QM predictions are not accurate enough to be trusted on their own
That is not entirely true: we are getting better and better at simulating things, and so often there's a reasonable chance that if you simulate a tiny bunch of molecules, their behaviour accurately matches real-life molecules.
However, we are not at the stage where people would take this for granted. Therefore, the ultimate test of whether a prediction is correct or wrong is to do the experiment in the lab. If the computation matches experiment, great: if not, then the computation is wrong. (Obviously, in this hypothetical and idealised discussion, we exclude unimportant considerations such as "the experimentalist messed up the reaction").
In a way, that means that you "can't predict chemistry": even if you could, it "doesn't count", because you'd have to then verify it by doing it in the lab.
(3) Whatever predictions we can make are too specific
There's another problem that is a bit more philosophical, but perhaps the most important. Let's say that we design a superquantum computer which allowed you to QM-simulate a gigantic bunch of molecules to predict how they would react. This simulation would give you an equally gigantic bunch of numbers: positions, velocities, orbital energies, etc. How would you distil all of this into a "principle" that is intuitive to a human reader, but at the same time doesn't compromise on any of the theoretical purity?
In fact, this is already pretty tough or even impossible for the things that we can simulate. There are plenty of papers out there that do QM calculations on very specific reactions, and they can tell you that so-and-so reacts with so-and-so because of this transition state and that orbital. But these are highly specialised analyses: they don't necessarily work for any of the billions of different molecules that may exist.
Now, the best you can do is to find a bunch of trends that work for a bunch of related molecules. For example, you could study a bunch of ketones and a bunch of Grignards, and you might realise a pattern in that they are pretty likely to form alcohols. You could even come up with an explanation in terms of the frontier orbitals: the C=O π* and the Grignard C–Mg σ.
But what we gain in simplicity, we lose in generality. That means that your heuristic cannot cover all of chemistry. What are we left with? A bunch of assorted rules for different use cases. And that's exactly what chemistry is. It just so happens that many of these things were discovered empirically before we could simulate them. As we find new theoretical tools, and as we expand our use of the tools we have, we continually find better and more solid explanations for these empirical observations.
Conclusion
Let me be clear: it is not true that chemistry is solely based on empirical data. There are plenty of well-founded theories (usually rooted in QM) that are capable of explaining a wide range of chemical reactivity: the Woodward–Hoffmann rules, for example. In fact, pretty much everything that you would learn in a chemistry degree can already be explained by some sort of theory, and indeed you would be taught these in a degree.
But, there is no (human-understandable) master principle in the same way that Newton's laws exist for classical mechanics, or Maxwell's equations for electromagnetism. The master principle is the Schrödinger equation, and in theory, all chemical reactivity stems from it. But due to the various issues discussed above, it cannot be used in any realistic sense to "predict" all of chemistry.

* Technically, this should be its relativistic cousins, such as the Dirac equation. But, let's keep it simple for now.
† In theory it cannot be solved for anything harder than a hydrogen atom, but in the last few decades or so we have made a lot of progress in finding approximate solutions to it, and that is what "solving" it refers to in this text.
