Is there a deep reason why action comes from a local lagrangian?
Lagrangian theories are indeed obiquitous because those are the ones we can understand better and ultimately those with which we can do computations. However, sadly, they only make sense when the theory admits a weakly coupled description. The reason is the very existence of the couplings. The fact that your theory has couplings means that there is a fine tuning that leads me to a free theory, where all the couplings are turned off. This is not a general property of QFTs. So there are two points that should be made
- Not all theories have a Lagrangian description in terms of weak interactions and weakly coupled fields.
- Those that have such a description must have an action functional of the form $S = \int \mathcal{L}$.
For point 1. there are several examples that can be constructed in supersymmetry for which one can prove that no Lagrangian description can exist. I am talking about $(2,0)$ six dimensional superconformal field theories. A non supersymmetric scenario which is unlikely but still viable is that of interacting seven dimensional conformal field theory (CFT). In that case it is obvious that no Lagrangian can exist since all interactions are irrelevant in 7d.
Non-Lagrangian constructions appear mostly in the realm of CFTs because, due to their rigidity, we can grasp their properties even if we essentially cannot compute anything. Suppose there exist two CFTs which are both strongly coupled and one flows into the other under renormalization group. Such a scenario would imply a QFT with no Lagrangian description anywhere in the energy range. And it is plausible to imagine that this might happen somewhat frequently.
A reference on the six dimensional CFT I mentioned is $[1]$. A paper where evidence for the existence of 7d CFTs is $[2]$.
As for 2. the reason is essentially locality. It is not known how to make a theory that respects the principle of microcausality if we start from an action that is not the integral of a local Lagrangian. In fact, it is believed that it is impossible to have a causal theory if the starting point is a nonlocal Lagrangian. By local I mean that $\mathcal{L}$ can be written as single spacetime integral of a polynomial of the fields at the same point with a finite number of derivatives. Sadly, I don't have references for this, perhaps some comments can improve this part of the discussion.
$[1]\;$ Christopher Beem, Madalena Lemos, Leonardo Rastelli, Balt C. van Rees The $(2,0)$ superconformal bootstrap 1507.05637;
See also Clay Cordova, Thomas T. Dumitrescu, Xi Yin Higher Derivative Terms, Toroidal Compactification, and Weyl Anomalies in Six-Dimensional $(2,0)$ Theories and refs. [10-16] therein. 1505.03850
$[2]\;$ Clay Cordova, G. Bruno De Luca, Alessandro Tomasiello $\mathrm{AdS}_8$ Solutions in Type II Supergravity 1811.06987
Dijkgraaf-Witten theories (Chern-Simons theory for a finite gauge group) are an example of a quantum theory without a Lagrangian, which nevertheless has an action (an exponentiated action to be precise). It is neither odd nor difficult to deal with, in fact, toy models like these are often used precisely because they are easier to handle than the quantum field theories we see in more familiar situations. The reason there is no Lagrangian here is because the gauge group is discrete, so there is no gauge field. Nevertheless, the quantum theory is perfectly well defined.
Lagrangians are particularly useful when we want to take a classical theory and get a quantum theory out of it, but if you look at any of the sets of axioms that go into defining a quantum field theory on its own, the Lagrangian does not really play much of a role.