This paper introduces new tests of fundamental physics by means of the analysis of disturbances on the GNSS signal propagation. We show how the GNSS signals are sensitive to a space variation of the fine structure constant $alpha$ in a generic framework of effective scalar field theories beyond the Standard Model. This effective variation may originate from the crossing of the RF signals with dark matter clumps and/or solitonic structures. At the macroscopic scale, the subsequent disturbances are equivalent to those which occur during the propagation in an inhomogeneous medium. We thus propose an interpretation of the measure of the vacuum permeability as a test of fundamental physics. We show the relevance of our approach by a first quantification of the expected signature in a simple model of a variation of $alpha$ according to a planar geometry. We use a test-bed model of domain walls for that purpose and focus on the measurable time delay in the GNSS signal carrier.
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