This Letter reports the first evidence of production of same-sign $W$ boson pairs where at least one of the $W$ bosons is longitudinally polarized and the most stringent constraint to date for the production of two longitudinally polarized same-sign $W$ bosons. The data set used corresponds to an integrated luminosity of 140 fb$^{-1}$ of proton-proton collisions at a center-of-mass energy of 13 TeV, collected with the ATLAS detector during Run 2 of the Large Hadron Collider. The study is performed in final states including two same-sign leptons (electrons or muons), missing transverse momentum, and at least two jets with a large invariant mass and a large rapidity difference. Two independent fits are performed targeting the production of same-sign $W$ bosons with at least one, or two longitudinally polarized $W$ bosons. The observed (expected) significance of the production with at least one longitudinally polarized $W$ boson is 3.3 (4.0) standard deviations. An observed (expected) 95% confidence level upper limit of 0.45 (0.70) fb is reported on the fiducial production cross section of two longitudinally polarized same-sign $W$ bosons.