This work investigates protograph-based lowdensity parity-check (LDPC) codes for the additive white Gaussian noise (AWGN) channel with on-off keying (OOK) modulation. A non-uniform distribution of the OOK modulation symbols is considered to improve the power efficiency especially for low signal-to-noise ratios (SNRs). To this end, a specific transmitter architecture based on time sharing is proposed that allows probabilistic shaping of (some) OOK modulation symbols. Tailored protograph-based LDPC code designs outperform standard schemes with uniform signaling and off-the-shelf codes by 1.1 dB for a transmission rate of 0.25 bits/channel use.