  Theories beyond the standard model involving a sub-GeV-scale vector $Z_d$
mediator have been largely studied as a possible explanation of the
experimental values of the muon and electron anomalous magnetic moments.
Motivated by the recent determination of the anomalous muon magnetic moment
performed at Fermilab, we derive the constraints on such a model obtained from
the magnetic moment determinations and the measurements of the proton and
cesium weak charge, $Q_W$, performed at low-energy transfer. In order to do so,
we revisit the determination of the cesium $Q_W$ from the atomic parity
violation experiment, which depends critically on the value of the average
neutron rms radius of $^{133}\text{Cs}$, by determining the latter from a
practically model-independent extrapolation from the recent average neutron rms
radius of $^{208}\text{Pb}$ performed by the PREX-2 Collaboration. From a
combined fit of all the aforementioned experimental results, we obtain rather
precise limits on the mass and the kinetic mixing parameter of the $Z_d$ boson,
namely $m_{Z_d} = 47{}^{+61}_{-16} \, \mathrm{MeV}$ and $\varepsilon =
2.3{}^{+1.1}_{-0.4} \times 10^{-3}$, when marginalizing over the $Z-Z_d$ mass
mixing parameter $\delta$.
