Defect-engineered and even amorphous two-dimensional (2D) materials have recently gained interest due to properties that differ from their pristine counterparts. Since these properties are highly sensitive to the exact atomic structure, it is crucial to be able to characterize them at atomic resolution over large areas. This is only possible when the imaging process is automated to reduce the time spent on manual imaging, which at the same time reduces the observer bias in selecting the imaged areas. Since the necessary datasets include at least hundreds if not thousands of images, the analysis process similarly needs to be automated. Here, we introduce disorder into graphene and monolayer hexagonal boron nitride (hBN) using low-energy argon ion irradiation, and characterize the resulting disordered structures using automated scanning transmission electron microscopy annular dark field imaging combined with convolutional neural network-based analysis techniques. We show that disorder manifests in these materials in a markedly different way, where graphene accommodates vacancy-type defects by transforming hexagonal carbon rings into other polygonal shapes, whereas in hBN the disorder is observed simply as vacant lattice sites with very little rearrangement of the remaining atoms. Correspondingly, in the case of graphene, the highest introduced disorder leads to an amorphous membrane, whereas in hBN, the highly defective lattice contains a large number of vacancies and small pores with no indication of amorphisation. Overall, our study demonstrates that combining automated imaging and image analysis is a powerful way to characterize the structure of disordered and amorphous 2D materials, while also illustrating some of the remaining shortcomings with this methodology.