First thing: all claims to have proved a causal relationship are relative to a set of assumptions which may or may not be satisfied (even in experimental sciences).
Taking the experiment as an ideal we cannot reach, there are many settings where there is useful variation in the data which can approximate the random assignment to treatment which is the essence of experiments.
Some canonical research designs which can be used here include differences in differences, regression discontinuity, synthetic control methods, or instrumental variables methods. They are differently appropriate to different settings but permit causal claims to be made. These methods are widely used in contemporary empirical economics (indeed this is a nearly exhaustive list of methods used in applied micro!).
Useful, accessible references here include Angrist and Pischke, “mostly harmless econométrics,” or “causal inference the mixtape” by a guy at Baylor whose name I’m forgetting, and the very recent “the effect” by Huntington-Klein, which I have not yet read.
Other slightly more exotic models are used in (for example) industrial organization which still permit causal claims to be made about the effects of increasing prices or changing product features on demand for a product or set of products.