Perturbation theory in Quantum/Computational Chemistry
Many people have already said this in the comments, but I will put it into an answer. Also, it would not be too hard to find a bunch of references in favor of how common any particular method is in quantum chemistry, so I won't bother to list a bunch of examples of perturbation theory being used.

That being said, if I had to guess, I would bet that second to DFT, second-order Moller-Plesset perturbation theory (MP2) is the most common method in quantum chemistry which includes electron correlation. Also, there are large communities in computational chemistry which virtually ignore the existence of DFT because they are willing to sacrifice computer-time (and possibly some accuracy) to be more certain that their answer is reliable. So, MP2 is a good example of perturbation theory being using in QM.

Also, you will often see people refer to coupled-cluster with singles and doubles excitations and a perturbative inclusion of the triples, CCSD(T), as the "gold-standard" in quantum chemistry. This is a bit more complicated application of perturbation theory, but I would say that CCSD(T) is used in virtually any problem where one can afford to do the calculation. There might be times where one could feel more confident about using DFT with a particular functional over MP2, but you would be hard-pressed to find a good argument to not do a CCSD(T) calculation if you can afford to do so. This is especially true if the alternative is to only use DFT because many functionals are parameterized against CCSD(T) calculations, so getting similar results from CCSD(T) helps validate the use of that functional for the class of problem.

There are also many other electronic structure methods which use perturbation theory such as CASPT2, one can also include perturbation excitations into truncated configuration interaction (CI) calculations, and probably lots of other but less-commonly-used methods. Also, one approach to including spin-orbit coupling in electronic structure is via perturbation theory.

One of the reasons there is such a vast array of methods which use perturbation theory is that they are diagrammatic theories, which means there is a rather simple way of understanding all of the terms physically and the connection between these terms and an algebraic representation of them is quite straightforward within the framework of second-quantization (basically using special creation and annihilation operators).

Also, I know you're mostly asking about electronic structure, but it should be noted that perturbation theory is often used to add anharmonicity onto a harmonic reference calculation of vibrational frequencies. Probably the most common method of doing this is via VPT2 which is basically the vibrational analogue of MP2.

In summary, perturbation theory is an extremely powerful tool which will almost certainly be applied to any problem where a simpler reference answer can be found. On this note, these days I don't think you could get away with doing only a Hartree-Fock calculation and expect people to believe your conclusions about the properties of the physical system. Also, there are such good, parallel codes for MP2 that if you can do a HF calculation, you can probably afford an MP2 calculation.



As a response to one of the comments, I would argue that CCSD by itself is not much more accurate than MP2 for the increased cost. Both methods can be understood as including double-excitations out of a reference determinant, and my experience is just doing CCSD does not improve much over MP2. Although MP2 and CCSD sometimes seem to be wrong in different ways, so they aren't strictly comparable. This is why people insist on CCSD(T) being the gold-standard and not just CCSD.