Interspacecraft ranging is crucial for the suppression of laser frequency noise via time-delay interferometry (TDI). So far, the effects of onboard delays and ambiguities on the Laser Interferometer Space Antenna (LISA) ranging observables were neglected in LISA modeling and data processing investigations. In reality, onboard delays cause offsets and timestamping delays in the LISA measurements, and pseudorandom noise (PRN) ranging is ambiguous, as it determines only the range up to an integer multiple of the PRN code length. In this article, we identify the four LISA ranging observables: PRN ranging, the sideband beat notes at the interspacecraft interferometer, TDI ranging, and ground-based observations. We derive their observation equations in the presence of onboard delays, noise, and ambiguities. We then propose a three-stage ranging sensor fusion to combine these observables in order to gain accurate and precise ranging estimates. We propose to calibrate the onboard delays on ground and to compensate the associated offsets and timestamping delays in an initial data treatment (stage 1). We identify the ranging-related routines, which need to run continuously during operation (stage 2) and implement them numerically. Essentially, this involves the reduction of ranging noise, for which we develop a Kalman filter combining the PRN ranging and the sideband beat notes. We further implement cross-checks for the PRN ranging ambiguities and offsets (stage 3). We show that both ground-based observations and TDI ranging can be used to resolve the PRN ranging ambiguities. Moreover, we apply TDI ranging to estimate the PRN ranging offsets.