Pulsar winds interacting with sources of external pressure are well-established as efficient and prolific TeV accelerators in our Galaxy. Yet, enabled by observations from Fermi-LAT, a growing class of non-accreting pulsars in binaries has emerged and these are likely to become apparent as TeV emitters in the CTA era. This class consists of the black widows and redbacks, binaries in which a millisecond pulsar interacts with its low-mass companion. In such systems, an intrabinary shock can form as a site of particle acceleration and associated nonthermal emission. We motivate why these sources are particularly interesting for understanding pulsar winds. We also describe our new multizone code which models the X-ray and gamma-ray synchrotron and inverse Compton spectral components for select spider binaries, including diffusion, convection, and radiative energy losses in an axially symmetric, steady-state approach. This new multizone code simultaneously yields energy-dependent light curves and orbital-phase-resolved spectra. It also better constrains the multiplicity of electron/positron pairs that have been accelerated up to TeV energies and are necessary to power orbitally-modulated synchrotron emission components between the X-rays and MeV/GeV bands potentially observed in some systems. This affords a more robust prediction of the expected high-energy and VHE gamma-ray flux. Nearby MSPs with hot or flaring companions may be promising targets for CTA, and it is possible that spider binaries could contribute to the observed AMS-02 energetic positron excess.