This Letter reports the first evidence of electroweak 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 dataset 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.