We present new constraints on three different models, the so-called universal, \(B-L\) and \(L_\mu-L_\tau\) models, involving a yet to be observed light vector \(Z'\) mediator, by exploiting the recent observation of coherent elastic neutrino-nucleus scattering (CE\(\nu\)NS) in argon and cesium-iodide performed by the COHERENT Collaboration. We compare the results obtained from a combination of the above data sets with the limits derived from searches in fixed target, accelerator, solar neutrino and reactor CE\(\nu\)NS experiments, and with the parameter region that could explain the anomalous magnetic moment of the muon. We show that for the universal and the \(B-L\) models, the COHERENT data allow us to put stringent limits in the light vector mediator mass, \(M_{Z'}\), and coupling, \(g_{Z'}\), parameter space.