According to the Lagrangian of the Standard Model, the g-factor of the W-boson is given at tree level by $g=2$.
Experimental results are compatible with this result, see eg p4 of https://arxiv.org/abs/hep-ex/0407042 for results from LEP.
There are radiative corrections to the tree level value which can be computed in the Standard Model: $g=2+\dots$, but, as far as I understand, the precision of the experiments is still insufficient to be sensible to these corrections.