A first measurement is presented of the cross section for the scattering of same-sign W boson pairs via the detection of a $\tau$ lepton. The data from proton-proton collisions at the center-of-mass energy of 13 TeV were collected by the CMS detector at the LHC, and correspond to an integrated luminosity of 138 fb$^{-1}$. Events were selected that contain two jets with large pseudorapidity and large invariant mass, one $\tau$ lepton, one light lepton (e or $\mu$), and significant missing transverse momentum. The measured cross section for electroweak same-sign WW scattering is 1.44$^{+0.63}_{-0.56}$ times the standard model prediction. In addition, a search is presented for the indirect effects of processes beyond the standard model via the effective field theory framework, in terms of dimension-6 and dimension-8 operators.