A search for Majorana neutrinos in same-sign $W^{\pm}W^{\pm}$ scattering events is presented. The analysis uses the proton-proton collision data recorded by the ATLAS detector at $\sqrt{s}= 13$ TeV at the Large Hadron Collider during the years 2015-2018, corresponding to an integrated luminosity of 140 fb$^{-1}$. The analysis targets final states including exactly two same-sign muons and at least two hadronic jets well separated in rapidity. The modelling of the main backgrounds from Standard Model, same-sign $W^{\pm}W^{\pm}$ scattering and $WZ$ production, are constrained with data in dedicated signal-depleted control regions and the transverse momenta of the sub-leading muon is used to search for signals originating from a heavy Majorana neutrino with masses of 50 GeV to 20 TeV. The presence of a signal is examined in a signal-enriched region and is interpreted in a benchmark scenario of the Phenomenological Type I Seesaw model. In addition, the sensitivity to the Weinberg operator is investigated. No significant excess is observed over the background expectation. Upper limits at the 95% confidence level are provided on the squared muon heavy neutrino mixing element $\vert V_{\mu N} \vert^{2}$ as a function of the heavy Majorana neutrino mass $m_N$, and on the effective $\mu\mu$ Majorana neutrino mass $|m_{\mu\mu}|$.