In recent years, a close connection between the description of open quantum systems, the input-output formalism of quantum optics, and continuous matrix product states (cMPS) in quantum field theory has been established. The latter constitute a variational class of one-dimensional quantum field states and have been shown to provide an efficient ansatz for performing tomography of open quantum systems. So far, however, the connection between cMPS and open quantum systems has not yet been developed for quantum transport experiments in the condensed-matter context. In this work, we first present an extension of the tomographic possibilities of cMPS by demonstrating the validity of reconstruction schemes based on low-order counting probabilities compared to previous schemes based on low-order correlation functions. We then show how fermionic quantum transport settings can be formulated within the cMPS framework. Our procedure, via the measurements of low-order correlation functions only, allows us to gain access to quantities that are not directly measurable with present technology. Emblematic examples are high-order correlations functions and waiting time distributions (WTD). The latter are of particular interest since they offer insights into short-time scale physics. We demonstrate the functioning of the method with actual data, opening up the way to accessing WTD within the quantum regime.