The first measurements of production cross sections of polarized same-sign $\mathrm{W}^{\pm}\mathrm{W^{\pm}}$ boson pairs in proton-proton collisions are reported. The measurements are based on a data sample collected with the CMS detector at the LHC at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of $137~\mathrm{fb}^{-1}$. Events are selected by requiring exactly two same-sign leptons, electrons or muons, moderate missing transverse momentum, and two jets with a large rapidity separation and a large dijet mass to enhance the contribution of same-sign $\mathrm{W}^{\pm}\mathrm{W^{\pm}}$ scattering events. An observed (expected) 95\% confidence level upper limit of 1.17 (0.88) fb is set on the production cross section for longitudinally polarized same-sign $\mathrm{W}^{\pm}\mathrm{W^{\pm}}$ boson pairs. The electroweak production of same-sign $\mathrm{W}^{\pm}\mathrm{W^{\pm}}$ boson pairs with at least one of the $\mathrm{W}$ bosons longitudinally polarized is measured with an observed (expected) significance of 2.3 (3.1) standard deviations.