Apparently correlation functions capture all the important information about a quantum field theory. Nonetheless, I have never been given a reason of why this should be the case. So, does anybody have any insight about why vacuum expectation values of operators should be so important?
From a practical point of view, the correlation functions are important because they are the quantities you need to determine to compute the scattering amplitudes by e.g. the LSZ reduction formula.
From an axiomatic point of view, the correlation functions are important because the Wightman reconstruction theorem says that, under certain assumptions, a Wightman QFT is fixed by its values for the n-point function, i.e. giving the complete set of n-point functions is equivalent to giving the actual fields.
Scattering amplitudes, transition probability, propagators and vertex amplitudes can be reduced to sum of products of correlations functions altogether (see the LSZ reduction formula, for example).