Can I install a water heater vent routed to the exterior wall which it is right next to?
It depends on the water heater design / type. Draft hood equipped water heaters (natural draft) have to vent above the roof, so technically you could go through the side wall, angle up, and continue above the roof. Power vented, condensing, and tankless WH can be vented through the side wall and terminate there unless otherwise specified in their installation requirements – that’s the best starting point. If you know the brand and model, go to the manufacturer’s website and search for installation instructions.