This dissertation presents two searches for supersymmetric particles performed with data collected from the ATLAS detector at the Large Hadron Collider. These searches are motivated by a supersymmetric model with an extra local symmetry $U(1)_{B-L}$. In this model the charged and neutral winos (superpartners to the Standard Model gauge bosons) or scalar top quarks may be the lightest supersymmetric particle and can decay through an $R$-parity violating coupling. The first search is presented for direct pair production of the stop,the supersymmetric partner of the top quark, in a decay through an $R$-parity violating coupling to a charged lepton and a $b$-quark. The dataset corresponds to an integrated luminosity of 140 $\mathrm{fb}^{-1}$ of proton--proton collisions at a center-of-mass energy of $\sqrt{s} = 13$~$\text{Te\kern -0.1em V}$ collected between 2015 and 2018 by the ATLAS detector at the Large Hadron Collider. The final state has two charged leptons (electrons or muons) and two $b$-jets. No significant excess is observed over the Standard Model background and exclusion limits on stop pair production are set at 95% confidence level. The corresponding lower limits on the stop mass for 100% branching ratios to a $b$-quark and an electron, muon, or tau-lepton are 1.9 TeV, 1.8 TeV and 800 GeV, respectively, extending the reach of previous Large Hadron Collider searches. The second search is for charged and neutral winos (charginos and neutralinos) using proton-proton collisions with the ATLAS detector at the Large Hadron Collider with a center-of-mass energy of $\sqrt{s} = 13$~$\text{Te\kern -0.1em V}$ collected between 2015 and 2018 and $\sqrt{s} = 13.6$~$\text{Te\kern -0.1em V}$ collected between 2022 and 2023. This search targets couplings of the chargino to muons or electrons, decays of the Higgs bosons to $b$-quarks, and events with one or two leptons in the final state. Work on the operation and calibration of the ATLAS Transition Radiation Tracker is also presented.