Hi wickidinhere,
If the solar panel is 10w, for say 10 hours of usable daylight, you could store ~100 watt-hours of usable power. So, the bug zapper wattage would have to be quite a pretty low to run several hours a night. I'm betting the zapper probably pulls at least 5A*12V=60W. So that would probably get you at most 2 hours a night.
I really enjoy dabbling in solar. I have a 45W panel, 55W controller, 1500W inverter, and 2 golf car batteries. This setup easily runs my tractor shed out in the sticks. I have a 2hp air compressor, a big shop vac, disk grinder, drill, light, etc. and I am never short on power. My system would probably run a 60W bug zapper ~7 hours a night. It cost ~$325 for the setup, less inverter.
