Electron-capture supernovae (ECSNe) may arise from ONeMg-core collapse in super-asymptotic giant branch (sAGB) stars near the low-mass core-collapse limit (≈8–10 M⊙). At early times, models predict that ECSNe resemble low-mass red supergiant iron-core-collapse SNe, making the two channels difficult to distinguish. Nebular spectroscopy, however, can reveal differences in ejecta composition. We present a systematic sample of 19 nebular spectra of low-luminosity Type IIP (LLIIP) SNe from the Zwicky Transient Facility Census of the Local Universe survey, obtained 115−450 days after explosion. Their low velocities expose narrow lines blended in brighter SNe, which we identify and model to constrain progenitor properties. We find a strong correlation between the FWHM of H i λ6563 and peak luminosity, showing that LLIIP SNe occupy the low-energy end of the core-collapse population, but no correlation with plateau duration, suggesting that envelope and core properties are not tightly linked. Only one SN reaches the extremely low H i λ6563 widths predicted for the weakest ∼9 M⊙ explosion models, implying that such low-energy events are intrinsically rare. Combining our sample with 118 literature nebular spectra of Type II SNe, we infer an IMF slope of 2.1 ± 1.2. We also introduce an “ECSN score” based on the absence of He- and O-shell emission lines, and identify two plausible ECSN candidates, SN 2023bvj and SN 2024btj. However, neither shows the extremely narrow nebular lines predicted by current ECSN models. If ECSNe arise predominantly through the LLIIP channel, we infer an upper limit on the ECSN rate of ≲(5–8) × 102 Gpc−3 yr−1, corresponding to a narrow sAGB progenitor mass window of ΔMsAGB ≲ 0.02–0.06 M⊙.